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CONIC SECTIONS.
ATiT
ELEMENTAKY TEEATISE
ON
CONIC SECTIONS
BY
CHARLES SMITH, M.A.,
MASTER OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE.
Uottbon I MACMILLAN AND CO.
AND NEW YORK. 1892
[The Right of Tramlation is reserved. ]
First Edition printed 1882. Second Edition printed 1883. Reprinted with slight corrections 1884, with slight cor- rections 1885, 1887, 1888, 1889, January and November 1890, 1892.
PEEFACE TO THE FIRST EDITION.
In the following work I have investigated the more elementary properties of the Ellipse, Parabola, and Hy- perbola, defined with reference to a focus and directrix, before considering the General Equation of the Second Degree. I believe that this arrangement is the best for beginners.
The examples in the body of each chapter are for the most part very easy applications of the book-work, and have been carefully selected and arranged to illustrate the principles of the subject. The examples at the end of each chapter are more difficult, and include very many of those which have been set in the recent University and College examinations, and in the examinations for Open Scholarships, in Cambridge.
The answers to the examples, together with occasional hints and solutions, are given in an appendix. I have also, in the body of the work, given complete solutions of some illustrative examples, which I hope will be found especially useful.
s. c. s. b
VI PREFACE.
Although I have endeavoured to present the ele- mentary parts of the subject in as simple a manner as possible for the benefit of beginners, I have tried to make the work in some degree complete; and have therefore included a chapter on Trilinear Co-ordinates, and short accounts of the methods of Reciprocation and Conical Pro- jection. For fuller information on these latter subjects the student should consult the works of Dr Salmon, Dr Ferrers, and Dr C. Taylor, to all of whom it will be seen that I am largely indebted.
I am indebted to several of my friends for their kind- ness in looking over the proof sheets, for help in the verification of the examples, and for valuable suggestions ; and it is hoped that few mistakes have escaped detection.
CHARLES SMITH.
Sidney Sussex College, April, 1882.
PREFACE TO THE SECOND EDITION.
The second edition has been carefully revised, and some additions have been made, particularly in the last Chapter.
Sidney Sussex College, July, 1883.
CONTENTS.
[The Articles marked with an asterisk may be omitted by beginners until after they have read Chapter IX.]
/<'
PAGE CHAPTER I. CO-OKDINATES 1
CHAPTER II. The Straight Line 12
Examples on Chapter II ...... . 43
CHAPTER III. Change of Axes. Anhakmonic Ratios or
Cross Ratios. Involution 47
CHAPTER IV. The Circle 60
Examples on Chapter IV 83
CHAPTER V. The Parabola 88
Examples on Chapter V 105
CHAPTER VI. The Ellipse 112
Examples, on Chapter VI 138
CHAPTER Vn. The Hyperbola ..'.... 146
Examples on Chapter VII 162
CHAPTER VIII. Polar Equation of a Conic, the focus being
the pole 166
Examples on Chapter VIII 176
CHAPTER IX. General Equation op the Second Degree.
Every curve whose equation is of the second degree is a conic 179
Co-ordinates of the centre of a conic 181
The Discriminant . 182
Position and magnitude of the axes of a central conic . . 183
Vlll • CONTENTS.
PAGE 184
185 188 189 190
Axis and latus rectum of a parabola Tracing conies .... Equation of the asymptotes of a conic Condition for a rectangular hyperbola Examples on Chapter IX
CHAPTEE X. Miscellaneous Pkopositions.
Equation of the tangent at any point of a conic . . . 192
Condition that a given straight line may touch a conic . 193
Equation of the polar of any point with respect to a conic . 194
Conjugate points and conjugate lines . . . . 195
A chord of a conic is cut harmonically by a point and its
polar 195
Diameters of a conic 196
Condition that two given lines may be parallel to conjugate
diameters ......... 196
Equi-conjugate diameters of a conic 197
Common conjugate diameters of two conies . . . 197
Segments of chords of a conic 198
Pairs of common chords of a circle and a conic are equally
inclined to the axes of the conic 199
Meaning ofS- \S'= 0, S -\uv=0 &nd S -\u^=0 . . 200
Equation of a pair of tangents 202
Equation of the director-circle 203
Four foci of a conic 204
Eccentricities of a conic 205
Equations giving the foci 207
Equation of the axes 209
Equation of a conic referred to tangent and normal . . 211
Normals 211
Similar conies 215
Examples on Chapter X . 219
CHAPTER XI. Systems of Conics.
One conic through five points 231
Conics through four points. [See also 295] . . . 233
Two parabolas through four points 234
Centre-locus of conics through four points. [See also 296] 234 Diagonal-points of a quadrangle are angular points of a triangle self-polar with respect to any circumscribing
conic. [See also 297.] 237
CONTENTS.
JDiagonals of a quadrilateral are sides of a triangle self-polar with respect to any inscribed conic ....
Centre-locus of conies touching four fixed lines. [See also 137, 270, and 297]
Parabola touching the axes of co-ordinates ....
Confocal conies
Osculating conies
Pairs of common chords of two conies
Invariants
Examples on Chapter XI
CHAPTER XII. Envelopes.
Envelopes, how found
Tangential co-ordinates and equations Director-circle of envelope (cf. 203)
Foci of envelope
Meaning of the tangential equation S - 'KS' = 0
Director-circles of conies which touch four given straight lines
have a common radical axis. [See also 308 and 319.] .
Examples
CHAPTEE Xin. Teilinear Co-obdinates.
Definition of Trilinear Co-ordinates . Equations and intersections of straight lines
The Kne at infinity
Parallel Unes
Condition of perpendicularity Length of perpendicular from a point on a line Co-ordinates of four points in the form ± /, ± gr, ± /i Equation of four lines in the form Za±m/3± 717=0
Examples
Tangent
Condition of tangency . Co-ordinates of the centre of a conic Condition for a parabola The asymptotes .... Condition for a rectangular hyperbola The circumscribing circle The circular points at infinity Conditions for a circle
IX
PAGE
238
240 241 244 251 253 254 257
264 267 268 269 269
270 271
276
278 280 280 281 282 283 284 285 286 287 288 288 289 290 290 291 291
X CONTENTS.
PAGE
Condition for ellipse, parabola, or hyperbola . . . 292
Foci 293
Equation of a circumscribing conic ..... 293
Equation of an inscribed conic . . . . . . 294
Equation of the inscribed circle 294
Conies through four fixed points 295
Conies touching four fixed lines 296
Conies with respect to which the triangle of reference is
self-polar 297
Conies referred to two tangents and the chord of contact . 298 The circle with respect to which the triangle of reference is
self-polar 298
The nine-point circle . 299
Pascal's theorem. [See also 328.] 299
Brianchon's theorem 301
Aeeal Co-ordinates 302
Tangential Co-ordinates . . . . . . . 303
Identical relation between the tangential co-ordinates of any
straight line 304
Tangential equation of a conic 305
The equation of the pole of a line 305
Examples on Chapter XIII 309
CHAPTEB XIV. Eeciprocal Polars. Projections.
Definition of polar reciprocal 315
The degree of a curve is the same as the class of its reciprocal 315
Examples of reciprocal theorems 317
Eeciprocation with respect to a circle 318
Beciprocal of a circle with respect to a point . . . 320
Co-axial circles reciprocated into confocal conies . , 322
Projection. Definition of projection 323
The projection of any curve is a curve of the same degree . 323
Projections of tangents, poles and polars .... 324 Straight lines meeting in a point projected into parallel
straight lines 324
Any line can be projected to infinity, and at the same time
any two angles into given angles 324
A quadrilateral projected into a square .... 326
Any conic projected into a circle 327
CONTENTS. XI
PAGE
A system of conies inscribed in a quadrilateral projected into
confocal conies 327
Cross ratios of pencils and ranges unaltered by projection . 329 Conies through four fixed points are cut by any straight line
in pairs of points in involution 329
The cross ratio of a pencil of four lines equal to that of the
range formed by their poles 329
Anharmonic properties of points on a conic, and of tangents
to a conic 330
nomographic ranges and pencils 331
Two lines at right angles together with the lines to the
circular points at infinity form a harmonic pencU . 334
Examples 336
Appendix 339
CHAPTEK I.
Co-ordinates.
1. If in a plane two fixed straight lines XOX\ YOY' be taken, and through any point P in the plane the two straight lines PM, PL be drawn parallel to XOX\ YOY respectively; the position of the point P can be found
when the lengths of the lines PM;PL are given. For we have only to take OL, OM equal respectively to the known lines PM, PL and complete the parallelogram LOMP.
The lengths MP and LP, or OL and OM, which thus define the position of the point P with reference to the
s. c. s. 1
2 CO-OKDINATES.
lines OX, OY are called the co-ordinates of the point P with reference to the axes OX, OY. The point of inter- section of the axes is called the origin. When the angle between the axes is a right angle the axes are said to be rectangular ; when the angle between the axes is not a right angle the axes are said to be oblique.
OL is generally called the abscissa, and LP the or- dinate of the point P.
The co-ordinate which is measured along the axis OX is denoted by the letter oo, and that measured along the axis 0 F by the letter y. If, in the figure, OL be a and OM be b ; then at the point P, x = a, and y = b, and the point is for shortness often called the point (a, h).
2. Let OM' be taken equal to OM, and OL' equal to OL, and through M', L' draw lines parallel to the axes, as in the figure to Art. 1. Then the co-ordinates of the three points Q, R, S will be equal in magnitude to those of P. Hence it is not sufficient to know the lengths of the lines OL, LP, we must also know the directions in which they are measured.
If lines measured in one direction be taken as positive, lines measured in the opposite direction must be taken as negative. We shall consider lines measured in the di- rections OX or OF to be positive, those therefore in the directions OX' or 0 Y must be considered negative.
We are now able to distinguish between the co-ordi- nates of the points P, Q, R, S. The co-ordinates of R are OL', L'R. and these are both measured in the negative direction ; so that, if the co-ordinates of P be a, b, those of R will be — a, — b. The co -ordinates of B will be a, — 6 ; and those of Q will be —a,b.
3. It must be carefully noticed that whether a line is positive or negative depends on the direction in which it is measured, and does not depend on the position of the origin ; for example, in the figure to Art. I, the line i/0 is negative although the line OL is positive.
If any two points K, L be taken and the distances
CO-OKDINATES. 3
OK, OL, measured from a point 0 in the line KL, be a and h respectively, then the distance KL must be K0+ OL, or — OK + OL, that is —a + h, and this will be the case wherever the point 0, from which distances are measured, may be.
If 0A=-^, and 05 = 4; then ^5= -(- 3) +4 = 7. If 0^ = 3, and 05= - 4 ; then ^J5= - 3 + ( - 4) = - 7.
The reader may illustrate this by means of a figure.
4. To express the distance between two points in terms of their co-ordinates.
Let P be the point (x, y), and Q the point {x', y"), and let the axes be inclined at an angle cd.
Draw FM, QL parallel to OY, and QR parallel to OX, as in the figure.
Then OL = x'\ LQ = y", OM = x', MP = y\ By trigonometry
PQ^ ==QE' + RP'-2QR. RP cos QRP. But QR = LM=^OM-OL = x-x",
RP = MP-MR = MP-LQ = y'-y", and angle QRP = angle OMP= ir — angle XO Y=7r-co, .-. PQ' = (x' - xy + (y - yy +.2{x' - x") {y' -y") cos a,, or PQ= ± ^J[{x'-x'y•^ {y'-y"y + 2(^' - x") {y' - y") cos o)}. If the axes be at right angles to one another we have PQ=±'J{{.x-x"f+{^-ff].
1—2
4 CO-ORDINATES.
The distance of P from the origin can be obtained from the above by putting x" — 0 and y" = 0. The result is
0P=± a/{(«" + y" + 2xy cos G)),
or, if the axes be rectangular.
Except in the case of straight lines parallel to one of the axes, no convention is made with regard to the direction which is to be considered positive. We may therefore suppose either PQ or QP to be positive. If however we have three or more points P, Q, R... in the same straight line, we must consider the same direction as positive throughout, so that in all cases we must have PQ+QE== PR.
5. To find the co-ordinates of a point which divides in a given ratio the straight line joining two given points.
Let the co-ordinates of P be w^, y^, and the co-ordi- nates of Q be ^2, 2/2, and let R (a;, y) be the point which divides PQ in the ratio k : I.
Draw PL, RN, QM parallel to the axis of y, and PBT parallel to the axis of x^ as in the figure.
Then LF : NM :: PB : ST :: PR:RQ::k: I;
or l{x — x^ -h{x^^ — x)=0;
CO-ORDINATES. 5
Similarly y Jl^^ ,
The most useful case is -when the line PQ is bisected : the co-ordinates of the point of bisection are
If the line were cut externally in the ratio k : I we should have
LN'.MNi'.kil, or LN'.NMv.k'.-l, •
and therefore x = — ,- , ^ , y = ~ — ^^^ . k-l ' ^ k-l
The above results are true whatever the angle between the co-ordinate axes may be. But in most cases formulae become more complicated when the axes are not at right angles to one another. We shall in future con- sider the axes to he at right angles in all cases except when the contrary is expressly stated.
Ex. 1. Mark in a figure the position of the point x = l, y = 2, and of the point £c=- 3, ?/=-!; and shew that the distance between them is 5.
Ex. 2. Find the lengths of the hnes joining the following pairs of points: (i) (1, - 1) and (- 1, 1) ; (ii) (a, -a) and (- &, 6) ; (iii) (3, 4) and ( - 1, 1).
Ex. 3. Shew that the three points (1, 1), { - 1, - 1) and ( - ^3, \/3), are the angular points of an equilateral triangle.
Ex. 4. Shew that the four points (0, -1), (-2, 3), (6, 7) and (8, 3) are the angular points of a rectangle.
Ex. 5. Mark in a figure the positions of the points (0, -1), (2, 1), (0, 3) and ( - 2, 1), and shew that they are at the corners of a square.
Shew the same of the points (2, 1), (4, 3), (2, 5) and (0, 3).
Ex. 6. Shew that the four points (2, 1), (5, 4), (4, 7) and (1, 4) are the angular points of a parallelogram.
Ex. 7. If the point (a;, y) be equidistant from the two points (3, 4) and (1, - 2), then will a; + 3i/ = 5.
CO-ORDINATES.
6. To express tJie area of a triangle in terms of the co-ordinates of its angular points.
Let the co-ordinates of the angular points A, B, C be ^i> 2/1; ^2> 2/2; and x^, ^3 respectively.
M X,
Draw the lines AK, BL, CM parallel to the axis of y, as in the figure.
A ABG = KAGM- MCBL - LBAK,
KAGM= A AGM-^ A ARM
= \KM.MG^\KM.KA
.-. A ABG =i[{y, + y,) {x, - x^j + (y^ + 2/,) (x, - x^)
+ (2/2 + ^1) K-^^J};
or, omitting the terms which cancel,
A ABG = J {x^y^ - x^^ -H x^^ - x^^ -f x^^ - x^y^}
Now
Similarly MGBL and LBAK
-\
|
^v |
Vv |
1 |
|
^2' |
2/2' |
1 |
|
X,, |
3/3> |
1 |
lattg]
The above expression for the area of a triangle will be found to be positive if the order of the angular points be such that in going round the triangle the area is always on the left hand. Whenever on substitution a negative result for the area is obtained, a reverse order of proceeding round the triangle has been adopted.
CO-ORDINATES.
7. To express the area of a quadrilateral in terms of the co-ordinates of its angular points.
Let the angular points A, B, G, D, taken in order, be
(^i> Vxl (^2. 2/2). (^3> ^3) and (x^, yj.
N M X
Draw AK, BL, GM, BN parallel to the axis of ?/, as in the figure.
Then the area ABGB
= KABL + LBGM- MGBN- NDAK.
And, as in the preceding Article,
KABL = 1 (y^ + y^ (x^ - x^),
LBGM=i{y^ + yX^s-^.)' MGDN=l{y^^y:){x,-x,),
. NBAK==l{y^ + y^){x,-x;). Hence ABGD = 1 {{y^ + ij^) (x^ - a^J + {y, + y,) (^, - ^\)
+ (^3 +2/4) (^4-^3) + (2/4 +.yi) (^1-^4)!;
or, omitting the terms which cancel, ABGD = J {y^^2- 2j,x^ + y,x^ - y,x^+ y,oo-y,x^ + y^a^i- y^ajj. The area of any polygon may be found in a similar manner.
Ex. 1. Find the area of the triangle whose angular points are (2, 1), (4, 3) and (2, 5).
Also find the area of the triangle whose angular points are (4, -5), (5, -6) and (3, 1). Ans. 4, |.
8 CO-ORDINATES.
Ex. 2. Find the area of the quadrilateral whose angular points are (1, 2), (3, 4), (5, 3) and (6, 2).
Also of the quadrilateral whose angular points are (2,2), ( - 2, 3), ( - 3, - 3) and (1, - 2). Ans. n., 20.
8. If a curve be defined geometrically by a property common to all points of it, there will be some algebraical relation which is satisfied by the co-ordinates of all points of the curve, and by the co-ordinates of no other points. This algebraical relation is called the equation of the curve.
Conversely all points whose co-ordinates satisfy a given algebraical equation lie on a curve which is called the locus of that equation.
For example, if a straight line be drawn parallel to the axis OY and at a distance a from it, the abscissae of points on this line are all equal to the constant quantity a, and the abscissa of no other point is equal to a.
Hence a;= a is the equation of the line.
Conversely the line drawn parallel to the axis of y and at a distance a from it is the locus of the equation x = a.
Again, if x, y be the co-ordinates of any point P on a circle whose centre is the origin 0 and whose radius is equal to c, the square of the distance OF will be equal to o^-Vy^ [Art. 4]. But OF is equal to the radius of the circle. Therefore the co-ordinates x, y of any point on the circle satisfy the relation c^ -\-y^ — (?. That is, x^ -\- y"^ = c* is the equation of the circle.
Conversely the locus of the equation x^ +y'^ — c^ is a circle whose centre is the origin and whose radius is equal to c.
In Analytical Geometry we have to find the equation which is satisfied by the co-ordinates of all the points on a curve which has been defined by some geometrical pro- perty ; and we have also to find the position and deduce the geometrical properties of a curve from the equation which is satisfied by the co-ordinates of all the points on it.
An equation is said to be of the w*^ degree when, after it has been so reduced that the indices of the vari-
CO-ORDINATES. 9
ables are the smallest possible integers, the term or terms of highest dimensions is of n dimensions. For example, the equations aooi/ + hx + c = 0, x^ + xy \Ja + lf = 0, and isjx + sjy = 1 are all of the second degree.
Ex. 1. A point moves so that its distances from the two points, (3, 4), and (5, - 2) are equal to one another ; find the equation of its locus.
Ans. a;-3i/ = l.
Ex. 2. A point moves so that the sum of the squares of its distances from the two fixed points (a, 0) and (-a, 0) is constant (2c2) ; find the equation of its locus. Ans. '3?--\-y'^ = (P'-a?'.
Ex. 3. A point moves so that the difference of the squares of its distances from the two fixed points (a, 0) and ( - a, 0) is constant (c^) ; find the equation of its locus. Ans. 4ax = ± c^.
Ex. 4. A point moves so that the ratio of its distances from two fixed points is constant ; find the equation of its locus.
Ex. 5. A point moves so that its distance from the axis of x is half its distance from the origin ; find the equation of its locus.
Ans. 3y2_x2 = o.
Ex. 6. A point moves so that its distance from the axis of x is equal to its distance from the point (1, 1) ; find the equation of its locus.
Ans. a;2-2«-2?/ + 2 = 0.
9. The position of a point on a plane can be defined by other methods besides the one described in Art. 1. A useful method is the following.
If an origin 0 be taken, and a fixed line OX be drawn through it ; the position of any point P will be known, if the angle XOF and the distance OP be given.
These are called the 'polar co-ordinates of the point P. The length OF is called the radius vector, and is
10 CO-ORDINATES.
usually denoted by r, and the angle XOP is called the vectorial angle, and is denoted by 6.
The angle is considered to be positive if measured from OX contrary to the direction in which the hands of a watch revolve.
The radius vector is considered positive if measured from 0 along the line bounding the vectorial angle, and negative if measured in the opposite direction.
If PO be produced to P', so that OP' is equal to OP in magnitude, and if the co-ordinates of P be r, 6, those of P' will be either r, ir -^ 6 ox — r, 6.
10. To find the distayice between two points whose polar co-ordinates are given.
Let the co-ordinates of the two points P, Qhe r^, 6^; and r^,d^.
Then, by Trigonometry,
PQ'=^OP'+ OQ' -20P . OQcos POQ.
But OP = r^, OQ = r^ a,nd A POQ=^XOQ-^ XOP=e.-e,)
PQ' = r^^ + r^ - 2r^r^ cos {6^ - 6>J.
The polar equation of a circle whose centre is at the point {a, a) and whose radius is c, is c'^ = a^ + i^ -2arcoB{6 - a)', where r, d are the polar co-ordinates of any point on it.
11. To find the area of a triangle having given the polar co-ordirmtes of its angular points.
o X
Let Pbe (r,, 6>J, Q be (r,, 6^), and R be {r^, 6^).
CO-ORDINATES.
n
Then area of triangle PQR = A POQ + A QOB- A FOR, and AFOQ=-iOP,OQ sin POQ
= J r^ r, sin (l9,-^ J, so AQ0R = ir^r^sm(6^-dX
and A P0i2= J ^3 r, sin ((93 - 0,)
.-. A PQR = i K ^. sin (0^ - e,) + r, r3 sin (^3 - 0,)
12. To change from rectangular to polar co-ordinates.
If through Oa line be drawn perpendicular to OX, and OX, OF be taken for axes of rectangular co-ordinates, we have at once
X = 0N= OP cos XOP= r cos 6, and y = NP== OP sin XOP= r sin 0.
Ex. 1. What are the rectangular co-ordinates of the points whose polar co-ordinates are (l,^j, (2, ^) and ( -4, - j j respectively?
Ex. 2. What are the polar co-ordinates of the points whose rect- angular co-ordinates are ( - 1, - 1), ( - 1, \/S) and (3, - 4) respectively?
Ex. 3. Find the distance between the points whose polar co-ordinates are (2, 40") and (4, lOO^) respectively.
Ex. 4. Find the area of the triangle the polar co-ordinates of whose
angular pomts are (1, 0), ( 1, ~\ and (\/2, jj respectively.
CHAPTER II.
The Straight Line.
13. To find the equation of a straight line parallel to one of the co-ordinate axes.
Let LP be a straight line parallel to the axis of x and meeting the axis oi y at L, and let OL = h.
Let X, y be the co-ordinates of any point P on the line. Then the ordinate iVPis equal to OL. Hence y = 6 is the equation of the line. Similarly x = a is the equation of a straight line parallel to the axis of y and at a distance a from it.
14. To find the equation of a straight line which passes through the origin.
Let OP be a straight line through the origin, and let the tangent of the angle XOP = m.
Let Xy y be the co-ordinates of any point P on the line.
THE STRAIGHT LINE.
13
Then Hence
i\rp = tan iV^OP. ON.
y — mx is the required equation.
Y
^ O NX
15. To find the eqtoation of any straight line.
Let LMP be the straight line meeting the axes in the points L, M.
Let OM^ c, and let tan OLM = 7n.
Let Xy y be the co-ordinates of any point P on the line.
Draw PN parallel to the axis of y, and OQ parallel to the line LMP, as in the figure.
Then NP = NQ-\~QP
= ONUnNOq + OM.
But NP = 2/, 0N= X, 0M= c, and tan ]S'OQ = tan OLM= m.
y = mx + c (i)
which is the required equation.
So long as we consider any particular straight line the quantities m and c remain the same, and are therefore called constants. Of these, m is the tangent of the angle
14 THE STRAIGHT LINE.
between the positive direction of the axis of x and the part of the line above the axis of x, and c is the intercept on the axis of y.
By giving suitable values to the constants 7/1 and c the equation y=mx-\-c may be made to represent any straight line whatever. For example, the straight line which cuts the axis of y at unit distance from the origin, and makes an angle of 45° with the axis of x^ has for equation y — x + \.
We see from (i) that the equation of any straight line is of the first degree.
16. To sheiu that every equation of the first degree i^epresents a straight line.
The most general form of the equation of the first degree is
Ax + By-\-G=^0 (i)
To prove that this equation represents a straight line, it is sufficient to shew that, if any three points on the locus be joined, the area of the triangle so formed will be zero.
Let {x\ y), {x'\ y"), and {x"\ y") be any three points on the locus, then the co-ordinates of these points will satisfy the equation (i).
We therefore have
Ax' -{-By + (7=0, Ax" +By" + G=0, Ax'"-\-By"'+G^(i. Eliminating A, B, G we obtain
X ,y' ,11=0, X ,y ,1\
the area of the triangle is therefore zero [Art. 6].
The equation Ax-\-By -vG — ^ is therefore the equa- tion of a straight line.
THE STKAIGHT LINE.
16
17. The equation Ax + By + G = 0 appears to in- volve three constants, whereas the equation found in Art. 15 only involves two. But if the co-ordinates x, y of any point satisfy the equation Ax -\- By + (7=0, they will also satisfy the equation when we multiply or divide through- out by any constant. If we divide by B, we can write
A G the equation 2/ = ~ p ^ "" d j ^"^^ ^^ h^ve only the two
A G
constants — ^ and — -^ which correspond to m and c in the
equation y = mx + c.
18. To find the equation of a straight line in terms of the intercepts which it makes on the axes.
Let A, B hQ the points where the straight line cuts the axes, and let OA — a, and OB = h.
Let the co-ordinates of any point P on the line be x,y.
Y
Draw PN parallel to the axis of y, and join OP, Then A APO + A PBO = A ABO ;
ay + hx = ah,
or
a b
This equation may be written in the form Ix 4- my = 1,
16
THE STRAIGHT LINE.
where I and m are the reciprocals of the intercepts on the axes.
19. To find the equation of a straight line in terms of the length of the perpendicular upon it from the origin and the angle which that perpendicular makes with an aads.
Let OL be the perpendicular upon the straight line AB, and let OL =p, and let the angle XOL = a.
Let the co-ordinates of any point P on the line be
Draw PJN' parallel to the axis of y, NM perpen- dicular to OL, and PK perpendicular to NM, as in the figure.
Then, OL = OM-\-ML = OM-h KP
= ON cos a + NP sin a ; or p = a) cos a + y sin a,
which is the required equation.
20. In Articles 15, 18 and 19 we have found, by independent methods, the equation of a straight line involving different constants. Any one form of the equation may however be deduced from any other.
For example, if we know the equation in terms of the intercepts on the axes, we can find the equation in terms of p and a from the relations a cos a = ^ and 6 sin a = p, which we obtain at once from the figure to Art. 19. Hence
THE STRAIGHT LINE.
17
substituting these values of a and b in the equation
- + 1 = 1, we get ic cos a + y sin a = p.
If the equation of a straight line be Ax-\-By-{- (7=0; then, by dividing throughout by JA^ -\- B^, we have A B G ^
Now
A
s/A^ + B"
and
s/A'+B' B
^A' + B' z^ are the cosine and sine
^/A' + B"
respectively of some angle, since the sum of their squares is equal to unity. If we call this angle a, we have occosa + y sin a —p = 0,
where p is put for , .
Ex. 1. If 3ic-4y-5 = 0, then dividing by Js^T¥ we have fa;-|y-l = 0. This is of the form a;cosa + 2/sina-_p = 0, where C08a = f, sin a = — ^, and p = l.
Ex. 2. The equation a; + y + 5=0, is equivalent to
Stt . Stt 6 X oos -r- + V sm -7- = — 7-
21. To find the position of a straight line whose equation is given, it is only necessary to find the co-
ordinates of any two points on it. To do this we may give s. c. s, 2
18 THE STRAIGHT LINE.
to X any two values whatever, and find from the given equation the two corresponding values of y. The points where the line cuts the axes are easiest to find.
Ex. 1. If the equation of a line be 2a: + 5?/ = 10. Where this cuts the axis of X, y=0, and then x = 5. Where it cuts the axis of y, x=0, and
y = 2.
Ex. 2. The intercepts made on the axes by the line Ax -y + 2=0 are - ^, and 2 respectively.
Ex. 3. x-2y = 0. Here the origin (0, 0) is on the line, and when x = 4:,y=2.
The lines are marked in the figure.
22. If we wish to find the equation of a straight line which satisfies any two conditions, we may take for its equation any one of the general forms.
(i) ^ = inx+c, (ii) - + |=1,
(iii) lx+7ni/= 1 , (iv) x cos a + y sin a —p = 0,
or (v) Ax + Bt/-{-G=0.
"We have then to determine the values of the two
constants m and c, or a and b, or I and m, or a and p,
A B
or -j^ and -^ for the line in question from the two con-
ditions which the line has to satisfy.
Ex. 1. Find the equation of a straight line which passes through the point (2, 3) and makes equal intercepts on the axes.
Let - + r = 1 be the equation of the line. a b
Then, since the intercepts are equal to one another, a = h.
Also, since the point (2, 3) is on the line,
? + ?=!;
a a
,'.a=5 = h and the equation required is v + f = !•
5 p
Ex. 2. Find the equation of the straight line which passes through
the point (Js, 2) and which makes an angle of 60^ with the axis of x.
hety=vix + c be the equation of the straight line.
THE STRAIGHT LINE. 19
Then, since the line makes an angle of 60'^ with the axis of x,
Also, if the point ( ,73, 2) be on the line, 2~m. /Js + c, therefore c = - 1, and the required equation is y= ^3x- 1.
23. To find the equation of a straight line drawn through a given point in a given direction.
Let x\ y' be the co-ordinates of the given point, and let the line make with the axis of x an angle tan"^ m.
Its equation will then be
y — mx + c, and, since {x, y) is on the line,
y = mx + c, therefore, by subtraction,
y-y'=-m(x-x') (i).
The line given by (i) passes through the point (x, y) whatever the value of m may be ; and by giving a suitable value to m the equation will represent any straight line through the point {x\ y).
If then we know that a straight line passes through a particular point {x ^ y') we at once write down y —y = m (x — x') for its equation, and find the value of m from the other condition that the line has to satisfy.
24. To find the equation of a straight line which parses through two given points.
Take any one of the general forms, for example,
y = mx + c (i).
Let the co-ordinates of the two points be x, y' and x'\ y" respectively. Then, since these points are on the line (i), we have
y' = mx' -tc (ii),
and 'if' = mx'-\- c (iii) .
From (i) and (ii), by subtraction,
y —y =m(x^x) (iv).
2—2
20
THE STRAIGHT LINE.
From (iii) and (ii), by subtraction,
y'-y = m(a;"-a;') (v)
y—y _ x—x
77 / Ti T •
y
and therefore
y X —X This equation could be found at once from a figure.
Ex. The equation of the line joining the points (2, 3) and (3, 1) is
25. Let the straight line AP make an angle 9 with the axis of x. Let the co-ordinates of A be x', y\ and those of P be ic, 3/, and let the distance -4P be r.
Draw AN, PM parallel to the axis of y, and AK parallel to the axis of x.
Then AK = ^Pcos (9, and KP = AP sin 6, or x^x =r cos 6, and y — y —t sin 6.
The equation of the line AP may be written in the form
x — x cos 6
y-y
sin 6?
r.
26.
Let the equation of any straight line be
Ax + By+ (7=0... (i).
Let the co-ordinates of any point Q be x\ y', and let the line through Q parallel to the axis of y cut the given straight line in the point P whose co-ordinates are x, y".
Then it is clear from a figure that, so long as Q remains on the same side of the straight line, QP is drawn in the same direction ; and that QP is drawn in the oppo- site direction, if Q be any point whatever on the other side of the straight line.
THE STEAIGHT LINE. 21
That is to say, QP is positive for all points on one side of the straight line, and negative for all points on the other side of the straight line.
Now QP^f-y' (ii),
and Aa^ + By-\-G=Ax' + By +G - {Ax' + By" + (7), [for {x\ y") is on the line, and therefore Ax + By" + (7 = 0] .-. Ax'+By'+C=-B{y"-y) (iii).
From (ii) and (iii) we see that Ax + By + (7 is positive for all points on one side of the straight line, and negative for all points on the other side of the line.
If the equation of a straight line be Ax + By 4-0=0, and the co-ordinates x , y of any point be substituted in the expression Ax -{-By + G; then if Ax' + By + (7 be positive, the point {x\ y') is said to be on the positive side of the line, and if Ax' + By' + (7 be negative, {x\ y') is said to be on the negative side of the line.
If the equation of the line be written
-Ax-By-G = 0,
it is clear that the side which we previously called the positive side we should now call the negative side.
Ex. 1. The point (3, 2) is on the negative side of 2x-Sy -1 = 0, and on the positive side of 3a; - 2y - 1 = 0.
Ex. 2. The points (2, - 1) and (1, 1) are on opposite sides of the line 3a; + 4y-6 = 0.
Ex. 3. Shew that the four points (0, 0), ( - 1, 1), ( - /^, 0) and (2, ^) are in the four different compartments made by the two straight lines 2x-3y + l = 0, and3a;-5t/ + 2 = 0.
27. To find the co-ordinates of the point of intersection of two given straight lines.
Let the equations of the lines be
ax -hby + c = 0 (i),
and ax + b'y + c=0 (ii).
Then the co-ordinates of the point which is common to both straight lines will satisfy both equations (i) and (ii).
22 THE STRAIGHT LINE.
We have therefore only to find the values of x and y which satisfy both (i) and (ii). These are given by
h& — b'c ca — ca ah' — a'h '
28. To find the condition that three straight lines may meet in a point
Let the equations of the three straight lines be
ow? + % + c = 0...(l), aa7 + 6'i/ + c' = 0...(2),
a"a; + 6"2/ + c" = 0...(3).
The three straight lines will meet in a point if the point of intersection of two of the lines is on the third.
The co-ordinates of the point of intersection of (1) and (2) are given by
a? ^ y ^ 1
he' — h'c ca' — c'a ah' — ah ' The condition that this point may be on (3) is „ he - h'c ,„ ca - c'a „ ab — ah ab —ab
or, a" (he - h'c) + b" (ca - c'a) + c" (ah' - ah) = 0.
EXAMPLES.
1. Draw the straight lines whose equations are
(i) x-\-y = 2, (ii) 3x-4y = 12,
(iii) 4x-3y+l = 0, and (iv) 2a; + 6y + 7=0.
2. Find the equations of the straight lines joining the following pairs of points— (i) (2, 3) and (-4, 1), (ii) (a, 6) and (6, a).
Am. (i) a:-3i/ + 7=0, (ii)a: + ?/ = a + 6.
i/ 3. Write down the equations of the straight lines which pass through
the point (1, - 1), and make angles of 160® and SO" respectively with the
axis of a;. * , ^ ^
^ns. y + l=T-^(x-l).
THE STRAIGHT LINE. 23
f/ 4 . Write the following equations in the form xcosa + ysma-p= 0, — (i) Sx + 'iy-15 = 0, (ii) 12a; -6y+ 10 = 0.
Ans. (i) 3^ + 1^-3 = 0, (ii) -nx + ^y-^ = 0.
1/ 6. Find the equation of the straight line through (4, 5) parallel to
2x-Sy-5 = 0. Ans. 2x-3y + 7 = 0.
j/ 6. Find the equation of the line through (2, 1) parallel to the line joining (2, 3) and (3, - 1). Ans. 4a; + y = 9.
7. Find the equation of the line through the point (5, 6) which makes equal intercepts on the axes, Ans. x + y = ll.
8. Find the points of intersection of the following pairs of straight
lines {i)5x + 7?/ = 99 and 3a; + 22/ + 77 = 0, (ii)2a;--5?/ + l = Oanda;+2/ + 4 = 0,
,..., X y ^ ^x y ^ (m)- + |.= land-+|=l.
Ans. (i) (-67, 62), (ii) (-3, -1), (iii) (-!^-^ , ±^ .
9. Shew that the three lines 5x + By-7 = 0, Sx-4y-10=0, and x + 2y=:0 meet in a point.
10. Shew that the three points (0, 11), (2, 3) and (3, - 1) are on a straight line.
Also the three points (3a, 0), (0, 36) and (a, 2b).
11. Find the equations of the sides of the triangle the co-ordinates of whose angular points are (1, 2), (2, 3) and (-3, -5).
Ans. 8x-5y-l = 0, 7x-4y + l = 0, x-y + l=0.
12. Find the equations of the straight lines each of which passes through one of the angular points and the middle point of the opposite side of the triangle in Ex. 11.
Ans. 2x-y=0, Sx-2y = 0, 5x-Sy = 0.
13. Find the equations of the diagonals of the parallelogram the equations of whose sides are x-a = 0, x- 6 = 0, y~c = (i and y -d = Q.
Ans. {d-c)x + {a-b)y + bc-ad=0 and(d-c) x + {b-a)y + ac-bd = 0.
14. What must be the value of a in order that the three lipes 3« + 2/-2 = 0, aa5 + 2y-3=0, and 2a;-y-3 = 0 may meet in a point?
Ans. a = 5.
15. In what ratio is the line joining the points (1, 2) and (4, 3) divided by the line joining (2, 3) and (4, 1)? Ans. The line is bisected.
16. Are the points (2, 3) and (3, 2) on the same or on opposite sides of the straight line 5y - 6x + 4 = 0 ?
24 THE STRAIGHT LINE.
17. Shew that the points (0, 0) and (3, 4) are on opposite sides of the line y-2«+ 1 = 0.
18. Shew that the origin is within the triangle the equations of whose sides area; — Ty + 25 = 0, 5a; + 3y + ll = 0, and Sx-2y-l = 0.
29. To find the angle between two straight lines whose equations are given.
(i) If the equations of the given lines be X cos a + 2/ sin a —p = 0, and x cos a' + y sin a —p = 0,
the required angle will be a — a or tt — a — a'.
For a and a! are the angles which the perpendiculars from the origin on the two lines respectively make with the axis of x, and the angle between any two lines is equal or supplementary to the angle between two lines perpen- dicular to them.
(ii) If the equations of the lines be
y = mx 4- c, and y = in'x + c ; then, if 6, & be the angles the lines make with the axis of £c, tan 6 = 7?^ and tan & = m\
.-. tan (^-(9)=—- 7;
^ 1 + mm
.'. the required angle is tan"^ ( z A .
The lines are perpendicular to one another when 1 + mm' = 0, and parallel when m = m' . (iii) If the equations of the lines be
ax + hy-{- c = 0, and a'x + h'y + c = 0, these equations may be written in the forms a c ^ a d
Therefore, by (ii), the required angle is
a a
^ ^i b b' ^ , ba — Ua
tan , or tan~^ — . ,,, .
- aa aa + 00
'^bb'
THE STRAIGHT LINE.
25
The lines aa)-{-hy + c = 0 and a'x + 6'y + c' = 0 will be at right angles to one another, if aa + hb' = 0, and
will be parallel to one another if ha — h'a = 0, or if — , = ^, .
30. The condition of perpendicularity is clearly satis- fied by the two lines whose equations are
ax + by + c=0 and hx — ay + G= 0. The condition is also satisfied by the two lines
ax + hy +c
X y
Oand--|+c' = 0. a 0
Hence if, in the equation of a given straight line, we
interchange, or invert, the coefficients of x and y, and alter
the sign of one of them, we shall obtain the equation of
a perpendicular straight line ; and if this line has to satisfy
some other condition we must give a suitable value to the
constant term.
Ex. 1. The line through the origin perpendicular to iy + 2x=7 is 2y-ix = 0.
Ex. 2. The line through the point (4, 5) perpendicular to 3a; - 2y + 5 = 0 is 2 (a; - 4) + 3(3/- 5) = 0, for it is perpendicular to the given line, and it passes through the point (4, 5).
Ex. 3. The acute angle between the lines
2x + By+l = 0, and x-y=0 is tan~i5.
31. To find the perpendicular distance of a given point from a given straight line.
Let the equation of the straight line be
xcosa + y sin a —p = 0 (i),
and let x', y' be the co-ordinates of the given point P.
26 THE STRAIGHT LINE.
The equation
a? cos a + ysma— ^' = 0 (ii)
is the equation of a straight line parallel to (i). It will pass through the point {x', y') if
X cosa ■\-y' sina— ^'= 0 (iii).
Now if PL be the perpendicular from P on the line (i), and ON, ON' the perpendiculars from the origin on the lines (i) and (ii) respectively, then will LP = NN'
= w cos a + y'sina—p [from (iii)].
Hence the length of the perpendicular from any point on the line xcos a + y sin a —p = 0 is obtained by substi- tuthig the co-ordinates of the point in the expression xcosa-^y sin a— p.
The expression x' cos a + 2/' sin a —^ is positive so long as p' is positive and greater than p, that is so long as P (x', y') and the origin are on opposite sides of the line.
If the equation of the line be aa; + 5y + c = 0, it may be written
x-\- -, 1/4— 7==0 (iv)
Va^ + 6' ^a^-Vb^ ^a' + b^ which is of the same form as (i) [Art. 20] ; therefore the length of the perpendicular from {x , y') on the line is a , h , c
'/drP'' 'J'^rPb^^ Va^TF'
ax +hy' +c , .
Va* + b' Hence, when the equation of a straight line is given in the form ax+by -\-c=0, the perpendicular distance of a given point from it is obtained by substituting the co-ordi- nates of the point in the expression ax + by + c, and dividing by the square root of the sum of the squares of the coeffi- cients ofx and y.
THE STRAIGHT LINE. 27
If the denominator of (v) be always supposed to be positive, the length of the perpendicular from any point on the positive side of the line will be positive, and the length of the perpendicular from any point on the negative side of the line will be negative. [See Art. 26.]
32. To find the equations of the lines which bisect the angles between two given straight lines.
The perpendiculars on two straight lines, drawn from any point on either of the lines bisecting the angles be- tween them, will clearly be equal to one another in mag- nitude.
Hence, if the equations of the lines be
aoo-hby + c = 0 (i),
and ax-^b'y + c =0 (ii),
and (x, y) be any point on either of the bisectors,
ax' -\-by' + c , ax + b'y + c
must be equal in magnitude.
Hence the point {x\ y') is on one or other of the
straight lines
ax-\-by + c , ax + b'y+c .....
—, ^_— = + ^ - (ni).
The two lines given by (iii) are therefore the required bisectors.
We can distinguish between the two bisectors; for, if we take the denominators to be both positive, and if the upper sign be taken in (iii), ax + by + c and ax -f b'y + c' must both be positive or both be negative.
-rj- . ax+by + c a'x + b'x -f c ,. \
Hence m - — . ^ = -\ ,— (iv),
every point is on the positive side of both the lines (i) and (ii), or on the negative side of both.
If the equations are so written that the constant terms are both positive, the origin is on the positive side of both
28 THE STRAIGHT LINE.
lines ; hence (iv) is the bisector of that angle in which the origin lies.
Ex. The bisectors of the angles between the lines ix-Sy + l=0, and
12x + oy + 13 = 0 are given by ^ = ± -^ ; and the upper
sign gives the bisector of the angle in which the origin lies.
33. To find the equation of a straight line through the point of intersection of two given straight lines.
The most obvious method of obtaining the required equation is to find the co-ordinates x, y of the point of intersection of the given lines; and then use the form y — y' =m (x — oc) for the equation of any straight line through the point {x, y). The following method is how- ever sometimes preferable. Let the equations of the two given straight lines be
ax -^ by -he = 0 (i),
a'x + h'y+c^O (ii).
Consider the equation
ax + hy-\-c + X {ax + b'y + c') = 0 (iii).
It is the equation of a straight line, since it is of the first degree; and if {x\ y) be the point which is common to the two given lines, we shall have
ax +by' â– ]-c=0 and oV + b'y' + c' = 0,
and therefore {ax' + by + c) +\ {a!x + b'y + c') = 0.
This last equation shews that the point (a;', y) is also on the line (iii).
Hence (iii) is the equation of a straight line passing through the point of intersection of the given lines. Also by giving a suitable value to \ the equation may be made to satisfy any other condition, it may for example be made to pass through any other given point. The equation (iii) therefore represents, for different values of \, all straight lines through the point of intersection of (i) and (ii).
THE STRAIGHT LINE. 29
Ex. Find the equation of the line joining the origin to the point of intersection of2x + 5y-4:=0 and 3x-2y + 2=6. Any line through the intersection is given by
2x + 5y-4: + \{Sx-2y + 2) = 0. This will pass through (0, 0) if -4 + 2\=0, or if X=2 ; .-. 2x + 5y-^ + 2{Sx-2y + 2)=0, or 8a; + 2/ = 0, is the required equation.
34. If the equations of three straight lines be ax + b^ + c = 0, aw + b'y + c' = 0, and a'x + b"y + c" = 0
respectively; and if we can find three constants \, fju, v such that the relation
\{ax + hy + c)-^ti {a'x + h'y -\-c')+v {a'oG+ h"y+c")=Qi...{i) is identically true, that is to say is true for all values of X and y, then the three straight lines will meet in a point. For if the co-ordinates of any point satisfy any two of the equations of the lines, the relation (i) shews that it will also satisfy the third equation. This principle is of fre- quent use.
Ex. The three straight lines joining the angular points of a triangle to the middle points of the opposite sides meet in a point.
Let the angular points A,B, C be [x', y'), {x", y"), {xf", y'"), respectively. Then D, E, F, the middle points of BG, CA, AB respectively, will be
^.'.^^ qr), (f::^, ti^y,^^^, t^y
The equation of AB will therefore be
y -y' x-x'
—2 y ~2 "^
or y{x" + x'" - 2x') - x {y" + y'" - 2y') + x' (y" + y'") - y' (a/' + x'") = 0.
So the equation of BE and CF will be respectively y {x'" + xf- 2x") - X {y'" + y' - 2y") + x" {y'" + y') - y" {x'" + a/) = 0 and y {x' + x" - 2x"') -x{y' + y" - 2y"') + x'" {y' + y") - y'" {x' + x") = 0.
And, since the three equations when added together vanish identically, the three lines represented by them must meet in a point.
30 THE STRAIGHT LINE.
EXAMPLES.
1. Find the angles between the following pairs of straight lines —
(i) y = 2x + 5 a.ndSx + y = 7,
(ii) x + 2y-4=0&nd2x-y + l = 0,
{iu)Ax + By + G=0 a.nd {A + B) x - {A - B)y = 0.
Alls, (i) 450, (ii) 900, (uj) 450.
2. Find the equation of the straight line which is perpendicular to 2x+7y-5 = 0 and which passes through the point (3, 1).
Am. 7x-2y = 19.
3. Write down the equations of the lines through the origin perpen- dicular to the lines Sx + 2y-5 = 0 and 4x + 3y-7 = 0. Find the co-ordinates of the points where these perpendiculars meet the lines, and shew that the equation of the line joining these points is 2305-1-11?/ - 35 = 0.
4. Find the perpendicular distances of the point (2, 3) from the lines 4a;-f3y-7-0, 5a;-f 12i/-20 = 0, and 3a:-|-4?/-8 = 0. Ans. 2.
5. Write down the equations of the lines through (1, 1) and ( - 2, - 1) parallel to 3a; -^ 42/4-7 = 0; and find the distance between these lines.
Ans. V-
6. Find the equations of the two straight lines through the point (2, 3) which make an angle of 45" with x + 2y=0.
Am. x-Sy + 7 = 0,dx + y = 9.
7. Find the equations of the two straight lines which are parallel to x + 7y + 2 = 0 and at unit distance from the point (1, - 1).
Am. x + 7y + 6±5j2=0.
8. Find the equation of the line joining the origin to the point of intersection of the lines x-'iy -7 = 0 and y + 2x-l=0.
Am. ldx + lly=0.
9. Find the equation of the straight line joining (1, 1) to the point of intersection of the lines 3a; -t- 4y - 2 = 0 and x-2y + 5=0.
Am. 7x + 26y-S3 = 0.
10. Find the equation of the line drawn through the point of inter- section of 2/ - 4a: - 1 = 0 and 2x + 5y-6=0, perpendicular to Sy + 4x=0.
A71S. 88t/- 66a; -101=0.
11. Find the lengths of the perpendiculars drawn from the origin on the sides of the triangle the co-ordinates of whose angular points are (2, 1), (3, 2) and (-1, -1).
12. Find the equations of the straight lines bisecting the angles between the straight lines 4:y + Sx-12 = 0 and 3t/ -f 4a; - 24 =0 ; and draw a figure representing the four straight Hues.
Am. 2/-a; + 12=0, 7y-l-7a;-36=0.
THE STRAIGHT LINE. 31
13. Find the equations of the diagonals of the rectangle formed by the lines x + 3i/-10 = 0, a;+3y-20=0, Sx-y + 5 = 0, and 3x-y-5=0; and shew that they intersect in the point (|, 1).
14. Find the area of the triangle formed by the lines y-x = 0, y-\-x=0, x-c = 0. Am. c^.
15. The area of the triangle formed by the straight lines whose equations are y - 2a; = 0, y - 3aj = 0, and y = 6a: + 4 is |.
16. Find the area of the triangle formed by the lines y = 2ic+4, 22/ + 3x = 5, and?/ + a: + l = 0. Ans. ^i^.
11. Shew that the area of the triangle formed by the lines whose equations are y^m-^x + c-^, y = m^-\-c^, and a; = 0 is
.(gi-g2)\ m^ — my '
18. Shew that the area of the triangle formed by the straight lines whose equations are 2/=%ic + Ci, y = in^ + C2, and y = m^x + c^ is
^{2^Z3l+:^ (Vi£i)!+ J {SlZ£2l, [Use Ex. 17.]
rua - Wg wig - THj wtj - Wj
19. Shew that the locus of a point which moves so that the sum of the perpendiculars let fall from it upon two given straight lines is constant is a straight line.
35. A homogeneous equation of the nth degree will represent n straight lines through the origin.
Let the equation be
Ay"" + %"-' X + a/-V' + . . . + Z"^" = 0. . .(i). Divide by a?" and we get
Let TWj, m^, mg m^ be the roots of this equation.
Then it is the same as
^(l-'':.)g-0(i-3) g-^^o.
and therefore is satisfied when
^- - m, = 0, when -^ - m„ = 0, &c., X ^ X ^
and in no other cases.
Therefore all the points on the locus represented by (i)
are on one or other of the n straight lines
y — m^x — ^, y — m^x = 0, y — m^x==0.
32 THE STRAIGHT LINE.
36. To find the angle between the two straight lines represented by the equation Ax^ + 2Bxy + C'lf = 0.
If the lines be y — m^x = 0, and y — m^x — 0, then (y — TWjjr) (2/ — m^x) = 0 is the same as the given equation
2/ + ^xy + -^x' = 0;
2J? ,.. , A ....
.. m^ + m^ = — -^ (1) sijid mjm^= jy (u).
If 6 be the angle between the lines,
tan e = ^b-^= ^^(f-^^) , from (i) and (ii).
If B^ — AG is positive the lines are real, being coin- cident if .B* - ulC = 0.
If B^ — AG \B> negative the lines are imaginary, but pass through the real point (0, 0).
The lines given by the equation Ax^+ 2l3xy 4- Gy'^= 0, will be at right angles to one another if A-\- C=0\ that is, if the sum of the coefficients of a? and y^ is zero.
37. To find the condition that the general equation of the second degree may represent two straight lines.
The most general form of the equation of the second degree is
aoj' + 2hxy + by' + 2gx -\-2fy-\-c = 0 (i).
If this is identically equivalent to
(Ix + my + n) [I'x + my â– \-n) = {) (ii),
we have, by equating coefficients in (i) and (ii), IV = a, mrn — b, nn — c, mn + m'n = 2/, nl' + nl = 2g, Im + I'm = 2h. By continued multiplication of the last three, we have 8fgh = 2ll'mm'nn + IV {m'^n^ + mV)
+ mm' {n'T + nH"") + nn' {V'm^ + IV) = 2ahc-ha{^f''-2bc)
+ b (4/ - 2ca) + c (4^^ - 2ab),
THE STRAIGHT LINE. 33
Hence ^ ahc - af - 6/ - dc" + '^fgh = 0 (iii)
is the required condition.
Unless the coefficients of a? and y^ are both zero, we can obtain the above result more simply by solving the equation as a quadratic in x or in y.
Suppose a is not zero ; then if we solve the quadratic in X, we have ax + hy + g=± V{(/t' - ah) y"" +^{hg - af)y -\- g"" - ac].
Now in order that this may be capable of being reduced to the form ax + By + C = 0, it is necessary and sufficient that the quantity under the radical should be a perfect square. The condition for this is
{V-ah)(<r-<w)={hg-af)\ which is equivalent to (iii).
38. To find the equation of the lines joining the origin to the common points of
ax^ + "ihxy + hy" + ^gx -\-2fy+c = 0 (i),
and Ix + my — 1 (ii).
Make equation (i) homogeneous and of the second degree by means of (ii), and we get ao^ + Ihxy + by^ + 2 (gx -\-fy) (Ix + my) + c (Ix + myY = 0
...(iii),
which is the equation required.
For equation (iii) being homogeneous represents straight lines through the origin [Art. 35]. To find where the lines (iii) are cut by the line (ii), we must put Ix + my = 1 in (iii), and we then have the relation (i) satisfied; which shews that the lines (iii) pass through the points common to (i) and (ii).
*39. To find the equation of the straight lines bisecting the angles between the two straight lines ax" + 2hxy + by^ = 0.
If the given lines make angles 0^ and 0^ with the axis of X, then \y — x tan d^ {y — ^ tan 6^ = 0 is the same as the given equation : and we obtain
s. c. S. 3
34 THE STRAIGHT LINE.
tan ^^ + taD ^2 = ~ X ^^^'
and tan^jtan^2 = y (ii).
If 6 be the angle that one of the bisectors makes with the axis of x, then will
and in either case
tan 26 = tan (6^ + 6^), 2tan^ _ tan ^^ 4- tan ^2 ^^ r^W'^^l-tan^jtanl?/
If {w, y) be any point on a bisector, -^ = tan 6)
X _ tan 0^ 4- tan 6^ hence "7""! -tan^^, tan 6»2 '
therefore, making use of (i) and (ii), we have for the re- quired equation
^xy _ 2h x^-f'a-h*
x^ — y^ xy
or f = -f .
a — b h
EXAMPLES.
1. Shew that the two straight lines \f-2xy^ecd + x^=0 make an angle d with one another.
2 . Shew that the equation x^ + xy- Qy^ â– \-7x + Sly -18 = 0 represents two straight lines, and find the angle between them. Ajis. 45*'.
3. Shew that each of the following equations represents a pair of straight lines, and find the angle between each pair:
(i) {x-a){y-a) = 0, (ii) x''-^y^ = 0,
(iii) xy^O, (iv) xy-2x-Sy + Q=0,
(v) a52-5xy + 4/ = 0, (vi) x^-5xy + ^y^+Sx-4:=^0,
(vii) x^ + 2xycot2a-y^ = 0,
THE STRAIGHT LINE. 35
4. For what value of X does the equation
12x*-10xi/ + 2y^ + llx-5tj + \ = 0 represent two straight lines? Shew that if the equation represents straight lines, the angle between them is tan~if. Aiis. X = 2.
5. For what value of X does the equation
12x^ + \xy + 2y^ + llx-5y + 2 = 0 represent two straight lines? Ans. - 10, or - »/.
6. For what value of X does the equation
12x^ + SQxy + \y^ + 6x + Qy+S = 0 represent two straight lines? Are the lines real or imaginary? Aiis. 28.
7. For what value of X does the equation '\xy + 5x + Sy + 2=0 represent two straight lines? Ans. \=\^.
8. Shew that the lines joining the origin to the points common to 3x^ + 5xy-^^ + 2x + Sy=0 and 3x-2y = l are at right angles.
The lines are 3x^ + 5xy - Sy^ + (2x+ Sy) {3x - 2y)=0.
Oblique Axes.
40. To find the equation of a straight line referred to axes inclined at an angle to.
Let LMP be any straight line meeting the axes in the points Z, M.
Let a?, y be the co-ordinates of any point P on the line.
Draw FN parallel to the axis of y and OQ parallel to the line LMP, as in the figure.
3—2
36 THE STRAIGHT LINE.
Then FP = NQ + QP (i)
^ ^ NQ sin NOQ ^ ^
-But -~r = -• — 7 — — wrw^ = coHstaiit = m suppose,
OJy sin (o) — JyOQ) ^^
and QP = OM = c suppose ;
therefore (i) becomes y = mx-\-c, which is the required
equation.
If 6 be the angle which the line makes with the axis
of Xy then
sin 6
sm {(a — By
^ m sin ft)
.'. tan 6 = .. .
1 -H m cos o)
41. Many of the investigations in the preceding Articles apply equally whether the axes are rectangular or oblique. These may be easily recognised.
*42. To find the angle between two straight lines whose equations, referred to axes inclined at an angle co, are given.
If the equations of the lines be
y = mx + c, and y = mw + c',
and if 0, 6' be the angles they make respectively with the
axis of X, then [Art. 40]
^ m sin o) i . /%/ ^' sin (o
tan V = tj , and tan 6 =
1 +m cos ft) 1 + m cos on '
ti r X /Zl nt^ (m — w') sin ft) ...
therefore tan (6 - 6) = -z — 7-^ jr-^ , . . . (1 ) ,
1 + (m + m)cosft) + m7n ^'
or the angle between the lines is
tan"' (^^ - mQ sin (o
1 + {m-\-m') cos ft) + mm! ' The lines will be at right angles to one another, if
1 + (m + m') cos ft) + wm' = 0 (ii).
If the equations of the two straight lines be ax+by + c = 0, and a'x + ¥y + c = 0,
and 0 be the angle between them, then m = —j, and
THE STRAIGHT LINE. 37
m' — — j-r, and substituting these values in (i) we have,
^ _ (ab — ah') sin co
aa + hb' — {ah' + a'h) cos cd *
The liues will be at right angles to one another, if aa + hb' — (ah' + a'b) cos o) = 0 (iii).
*43. To find the perpendicular distance of any point (/, g) from the line Ax + By+G=0.
Let the line cut the axes of x and y m the points K, L respectively, and let P be the point whose co-ordinates are /, g, and let PN be the perpendicular from it on the line LK. Then
APLK=APOK-{-APLO-ALOK (i),
.-. PK LK=OK.gsm(o+ OL ./sin a> - OK. OL sin co...
(ii);
The relation expressed in (i) requires to be modified for different positions of the point and of the line, unless we make some convention with respect to the sign of the area of a triangle, but the equation (ii) is universally true. The student should convince himself of the truth of this by drawing different figures.
Now ^^ = "3' ^^ = -|'
also LIP = OIC +0L'-2 OK . OL cos co
= -2^2 (^' + ^"- 2-^^ cos o)) ;
„ ...^ ^,^ Af+Bg+C .*. from (ii) PiY= -;r-«2"n2 — Sn^ r^m oo.
*44. To find the angle between the lines ax'' + 2hxy + by"" = 0, the axes being inclined at an angle ay. If the lines be y=m'x and y^mx^
, „ 1h then will m + m =—-r- ,
38 THE STRAIGHT LINE.
and mm =r;
, , „ 2 Jh — ah
whence m — m = ^ ^ .
b
But the angle between y = m'x and y = m"x is
tan-.^'^,!"!^''^.,-^, [Art. 42]; 1 + (m + m ) cos (0 + mm _ â– "
therefore the angle required is
_i 2 \/{h^ — ah] sin &>
tan —J — -^ .
0 — Zfi cos Q}-\- a
The lines aa;'^ + 2hxy + by^ = 0 are at right angles t6 one another, if
a+b — 2h cos (0 = 0.
Polar Co-ordinates.
45. To find the polar equation of a straight line.
Let 01^ be the perpendicular on the given line from the origin, and let ON = p, and XON= a.
Let P be any point on the line, and let the co-ordinates of P be r, e.
Then, in the figure, z NOP is (0 — a), and
OPcos NOP = ON. Therefore the required equation is r cos {0 — a) =p. This equation may also be obtained by writing r cos 0 for x, and r sin 0 for y in the equation ^ cos a + y sin a = p.
THE STRAIGHT LINE. 39
46. To find the polar equation of the line through two given points.
Let P, Q be the given points and let their co-ordinates be /, 6' and r'\ &' respectively.
Let R be any other point on the line, and let the co-ordinates of jR be (r, &). Then, since
A PO (3 -t- A qOR - A FOE = 0, we have
/ r' sin {0"-e') + r''r sin {0 - 6") - rr sin [6 - 6') = 0.
The required equation is therefore, rr" sin [6" - ff) -\- r"r sin {6 - 6") + rr' sin {6' - ^) = 0.
EXAMPLES.
1. Shew that the lines given by the equation i/-x'^ = (i are at right angles to one another whatever the angle between the axes may be.
2. Find the equation of the straight line passing through the point (1, 2) and cutting the line x + 2y = 0 at right angles, the axes being inclined at an angle of 60*'. Ans. x=l.
3. Find the angle the straight line y = ox + ^ makes with the axis of X, the axes being inclined at an angle whose cosine is |. Ans. 45<'.
4. li y=7nx + c and y^mx + c' make equal angles with the axis of as, then will m + w' + 1mm' cos w = 0.
6. If the lines Ax^ + 2Bxy + Cy'^ = Q make equal angles with the axis of x, then will B = A cos w.
6. Shew that the lines given by the equation
£c2 + 2a;?/cos w + y"^ cos 2w = 0 are at right angles to one another, the axes being inclined at an angle w.
7. Find the polar co-ordinates of the foot of the perpendicular from the pole on the line joining the two points (r^ d^y {r^, d^.
47. We shall conclude this chapter by the solution of some examples.
(1) On the sides of a triangle as diagonals, parallelograms are described, having their sides parallel to trco given straight lines ; shew tliat the other diagonals of these parallelograms will meet in a point.
40
THE STRAIGHT LINE.
Take any two lines parallel to the sides of the parallelograms for the axes. Let A, B, C, the angular points of the triangle, be (x', y'), {x", y") and {x'", y'") respectively.
Then the extremities of the other diagonal of the parallelogram of which AB is, one diagonal will be seen to be [x', y") and (.r", y'). Therefore the equation of the diagonal FK will be y -y" _x -TJ \f-y"-^'-^^ or X {y' - y") +y{x'- x") + x"y" - x'y' = 0.
Similarly the equation of HE will be
X {y" - y'") + y [x" - x'") + x"'y"' - x!'y" = 0, and the equation of GD will be
X [y'" -y')+y {^"' - .-^0 + ^y - x'y = o.
The sum of these three equations vanishes identically, therefore the three straight lines meet in a point. [Art. 34.]
(2) Any straight line is drawn through a fixed point A cutting two given straight lines OX, OY in the points P, Q respectively, and the paral- lelogram OPRQ is completed: find the equation of the locus o/R.
THE STRAIGHT LINE. 41
Take the two given lines for the axes, and let the co-ordinates of the point A be /, g.
Let the equation of the line PQ, in any one of its possible positions be
^^=^ »•
Then the co-ordinates of the point R will be a and /3.
But, since the line PQ passes through the point (/, ^), the values aj=/, ?/ = <7 satisfy the equation (i). Therefore
M=i '")•
Hence the co-ordinates a and ^ of the point R always satisfy the relation (ii). Calling the co-ordinates of the point JR, x and y instead of a and j8, we have for the equation of its locus
X y
(3) Through a fixed point 0 any straight line is drawn meeting two given parallel straight lines in P and Q ; through P and Q straight lines are drawn in fixed directions, meeting in R: prove that the locus of B, is a straight line.
Take the fixed point O for origin, and the axis of y parallel to the two parallel straight lines; and let the equations of these parallel lines be x=a, x — h.
Then, if the equation of OPQ, be y = mx, the abscissa of P is a, and therefore its ordinate ma ; also the abscissa of Q is &, and therefore its ordinate mb.
Let PR be always parallel to y = m'x and QR always parallel ioy = m"x, then the equation of PR will be
y - ma = vi' {x- a) (i),
and the equation of QR will be
y -vib = m" {x-b) (ii).
At the point R the relations (i) and (ii) will both hold, and we can find, for any particular value of m, the co-ordinates of the point R by solving the simultaneous equations (i) and (ii). This however is not what we want. What we require is the algebraic relation which is satisfied by the co-ordinates {x, y) of the point R, whatever the value of m may be. To find this we have only to eliminate m between the equation (i) and (ii).
42 THE STRAIGHT LINE.
The result is
{h-a) y = m'b (x-a)- m"a {x-h).
This equation is of the first degree, and therefore the required locus is a straight line.
(4) To find the centres of the inscribed circle and of the escribed circles of a triangle whose angular points are given.
Let (x', y'), {x", y"), {x"\ y'") be the angular points A, B, C respectively. The equation of 5 C is
y{x"-x"')-x{y"-y'") + y''x'"-x'Y'=0 (i).
the equation of CA is
yix'"-x')-x{y"'-y')i-y"'x'-x"Y-^0 (ii),
and the equation of AB is
y{x'-x")~x{y'-y'^ + y'x"-x'y" = 0 (iii).
The perpendiculars on these lines from the centre of any one of the circles are equal in magnitude.
The centres of the four circles are therefore [Art. 31] given by ^ y {x" - a^'-Q -xjt/'- y'") + y"x"' - ^'f' J{x"-x"r + {3/'-fy ^^y {^" -x')-x {f - if) + f'x' - x!'Y
sl{x"'-xy + {f'-yr ^ ^ y{^-x")-x (y'j-y'J+y'x" - x'y"
Ji^-xy+^if-yy
If the co-ordinates of the angular points A, B, C of the triangle be substituted in the equations (i), (ii), (iii) respectively, the left hand mem- bers of all three will be the same. Hence, [Art. 26] the angular points of the triangle are either all on the positive sides of the lines (i), (ii), (iii), or all on the negative sides.
The perpendiculars from the centre of the inscribed circle on the sides of the triangle are all drawn in the same direction as those from the angular points of the triangle. Hence in (iv) the signs of all the ambiguities are positive for the inscribed circle.
For the escribed circles the signs are - -f -f , + - 4- , and + + - respectively.
THE STRAIGHT LINE. 43
Examples on Chapter II.
1. A straight line moves so that the sum of the recipro- cals of its intercepts on two fixed intersecting lines is constant ; shew that it passes through a fixed point.
2. Prove that hv^ — Ihxy + ay^ = 0 represents two straight lines at right angles respectively to the straight lines
ax^ + 2hxy + hy^ = 0.
3. Find the equation to the n straight lines through (a, h) perpendicular respectively to the lines given by the equation
4. Find the angles between the straight lines represented by the equation
5. OA, OB are two fixed straight lines, A, B being fixed points; P, Q are any two points on these lines such that the ratio oi AP to BQ is constant; shew that the locus of the middle point of PQ is a straight line.
6. If a straight line be such that the sum of the perpendi- culars upon it from any number of fixed points is zero, shew that it will pass through a fixed point.
7. PM, PN are the perpendiculars from a point P on two fixed straight lines which meet in 0 ; MQ, NQ are drawn parallel to the fixed straight lines to meet in Q ; prove that, if the locus of P be a straight line, the locus of Q will also be a straight line.
8. A straight line OPQ is drawn through a fixed point 0, meeting two fixed straight lines in the points P, Q, and in the straight line OPQ a point R is taken such that OP, OR, OQ are in harmonic progression; shew that the locus of P is a straight line.
9. Find the equations of the diagonals of the parallelogram formed by the lines
a = 0, a = c, a =:0, a'=c, where a = re cos a + y sin a-p,
and a =x cos a! + y sin a —p'.
44 THE STRAIGHT LINE.
10. ABCD is a parallelogram. Taking A as pole, and AB as initial line, find the polar equations of the four sides and of the two diagonals.
11. From a given point {h, k) pei-pendiculars are drawn to the axes and their feet are joined; prove that the length of the perpendicular from (A, k) upon this line is
hk sin^o) ^{A'4-^+2Mcosa>}' and that its equation is hx-ky = h^ — k^.
12. The distance of a point (ajj, ?/,) from each of two straight lines, which pass through the origin of co-ordinates, is S; prove that the two lines are given by
13. Shew that the lines FC, KB^ and ^Z in the figure to Euclid I. 47 meet in a point.
14. Find the equations of the sides of a square tlie co-ordinates of two opposite angular points of which are 3, 4 and 1, - 1.
15. Find the equation of the locus of the vertex of a triangle which has a given base and given difference of base angles.
16. Find the equation of the locus of a point at which two given portions of the same straight line subtend equal angles.
17. The product of the perpendiculars drawn from a point on the lines
X cos 6 -^y sin 6= a, x cos <f> + y sin <j> = a
is equal to the square of the perpendicular drawn from the same point on the line
0 + d> . e + d> e-di
X cos — ^ + y sm —~- ^ a cos — — - ;
Ji Z Z
shew that the equation of the locus of the point is x^ + y* — a*.
18. PA, PB are straight lines passing through the fixed points A^ B and intercepting a constant length on a given straight line ; find the equation of the locus of P.
THE STRAIGHT LINE. 45
19. The area of the parallelogram formed by the straight lines 3x + 4:]/ = 7c*j, Sx + 4:y= 7a^, 4:X + 3t/^ 7by, and ix + 3^/= 7b^ is 7 K- a,) (6,-6,).
20. Shew that the area of the triangle formed by the lines aa^ + Vixy + hif = 0 and Ix + my + n = 0 is
aw? - 2hlm + hl^'
21. Shew that the angle between one of the lines given by aa? + 2hxy + hy^ =^ 0, and one of the lines
ax^ + '2^kxy + by^ + \ {x^ + y^) = 0, is equal to the angle between the other two lines of the system.
22. Find the condition that one of the lines
ax^ + 2hxy + hy^ = 0, may coincide with one of the lines
a'x^ + Ihixy + V'lf = 0.
23. Find the condition that one of the lines
ao(? + ^hxy + hy^ = 0, may be perpendicular to one of the lines aV + 2h'xy + b'y^ = 0.
24. Shew that the point (1, 8) is the centre of the inscribed circle of the triangle the equations of whose sides are
4:y + 3x = 0, 12y - 5x = 0, y- 15=^0, respectively.
25. Shew that the co-ordinates of the centre of the circle inscribed in the triangle the co-ordinates of whose angular points are(l, 2), (2, 3) and (3, 1) are ^(8 + 710) and 1(16-^^10). Find also the centres of the escribed circles distinguishing the different cases.
26. If the axes be rectangular, prove that the equation
{x'-3f)x = 7ny(i/'-3x') represents three straight lines through the origin making equal angles with one another.
27. Shew that the product of the perpendiculars from the point (x'y y) on the lines aaj^ -f- 2hxy 4- by^ = 0, is equal to
ax'' + 2hx'y + by'' J{a-by + W
46 THE STRAIGHT LINE.
28. If ^j, jt?g be the perpendiculars from (cc, y) on the straight lines ax^ + 2hxy + hy^ = 0, prove that
+ 4:h{a + h) xy + ih^ {x^ + y^).
29. Shew that the locus of a point such that the product of the perpendiculars from it upon the three straight lines represented by
ay^ 4- hy^x + cyx^ + dx^ = 0 is constant and equal to k^ is ay^ + hy'^x + cyx^ + dx^ - ¥ J{a-cY + {p-df= 0.
30. Shew that the condition that two of the lines re- presented by the equation
Ax"" + 2,Bx'y + ZCxy' + Dy'=0 may be at right angles is
31. Shew that the equation
a {x* + y*) - ibxy (of - y") + 6cxy = 0 represents two pairs of straight lines at right angles, and that, if 2b' = a' + Sac, the two pairs will coincide.
32. The necessary and sufficient condition that two of the lines represented by the equation
ay* + bxy^ + cx'y' + dx^y + ea;* = 0 should be at right angles is
(6 + c? ) (ao? + be) + (e - a)* (a + c + e) = 0.
33. Shew that the straight lines joining the origin to the points of intersection of the two curves
ax' + 2hxy + b^ + 2gx = 0, and a V + 'Ih'xy + b'y' + 2g'x = 0,
will be at right angles to one another, if g\a + b) =g (a' + 6').
34. Prove that, if the perpendiculars from the angular points of one triangle upon the sides of a second meet in a point, the perpendiculars from the angular points of the second on the sides of the first will also meet in a point.
35. If the angular points of a triangle lie on three fixed straight lines which meet in a point, and two of the sides pass through fixed points, then will the third side also pass through a fixed point.
CHAPTER III.
Change of Axes. Anharmonic Ratios, or Cross Ratios. Involution.
Change of Axes.
48. When we know the equation of a curve referred to one set of axes, we can deduce the equation referred to another set of axes.
49. To change the origin of co-ordinates without changing the direction of the axes.
Let OX, OF be the original axes; O'X', O'Y' the new axes; O'X' being parallel to OX, and O'Y being parallel to OY. Let /i, k be the co-ordinates of 0' referred to the original axes,
48
CHANGE OF AXES.
Let P be any point whose co-ordinates referred to the old axes are x, y, and referred to the new axes x\ y , Draw FM parallel to OY, cutting OX in 1/ and 0' X' inK
Then x = OM = OK + KM= OK + ON =h+x, y = MP = MN+NP = KO' + XP = k + y'. Hence the old co-ordinates of any point are found in terms of the new co-ordinates ; and if these values be substituted in the given equation, the new equation of the curve will be obtained.
In the above the axes may be rectangular or oblique.
50. To change the direction of the axes without changing the origin, both systems being rectangular.
|
kP |
|||||
|
y\ |
Y |
\ |
|||
|
L |
\ |
^ |
|||
|
^^ |
n' |
||||
|
0 |
1 |
T 1 |
d |
X |
Let OX, 0 r be the original axes ; OX', OY the new axes ; and let the angle XOX = 6.
Let P be any point whose co-ordinates are x, y re- ferred to the original axes, and x , y referred to the new axes. Draw PN perpendicular to OX, PN perpendicular to 0X\ N'M perpendicular to OX, and N'L perpendicular to PN, as in the figure. Then x = ON=OM-NM=^OM- LN' = 0N' cos e-X'P sine = x cos d — ys\nQ\ y = NP=NL-\-LP = MX' + LP = ON' sin e + NP cos 6 = x sin 0-\-y cos 6.
CHANGE OF AXES. 49
Hence the old co-ordinates of any point are found in terms of the new co-ordinates; and if these vahies be substituted in the given equation, the new equation of the curve will be obtained.
Ex. 1. What does the equation Sx^ + 2xy + 3y- - 18a; - 22y + 50 = 0 he- come when referred to rectangular axes through the point (2, 3), the new axis of X making an angle of 45" with the old?
First change the origin, by putting x' + 2, ?/' + 3 for x, y respectively. The new equation will be 3(x' + 2)2 + 2(x' + 2)(y' + 3) + 3(y' + 3)2-18(x' + 2)-22(y' + 3) + 50 = 0; which reduces to 3x'^ + 2x'y' + 3y '^ - 1 = 0,
or, suppressing the accents, to
3x2 + 2«r/ + 37/2 = 1 (i^
To turn the axes through an angle of 45" we must write oi -~--y' —
\/2 \J2
for a;, and x' — r- + y' ——. for y. Equation (i) will then be
which reduces to 4x'^ + 2y'^ = l.
Ex. 2. What does the equation x^ -y^ + 2x + 4:y = 0 become when the origin is transferred to the point (- 1, 2)? Ans. x^-y^+S=0.
Ex. 3. Shew that the equation 6^2 ^ 5^^ _ Qy2 _ j y^j + 7^ ^ 5 _ 0, when referred to axes through a certain point parallel to the original axes will become 6x2 4. ^^y _ g^s _ q.
Ex. 4. What does the equation 4x2 + 2>/3x7/ + 2y^-l = 0 become when the axes are turned through an angle of 30"? Ans. 5x^ + y^ -1 = 0.
Ex. 5. Transform the equation x^~2xy + y^ + x-3y = 0 to axes through the point (-1, 0) parallel to the lines bisecting the angles be- tween the original axes. Ans. \/2y'^ — x = 0.
Ex. 6. Transform the equation x^ + cxy + y^ = a^, by turning the rect- angular axes through the angle - .
51. To change from one set of oblique axes to another, xuithout changing the origin.
S. C. S. 4
50
CO
CHANGE OF AXES.
Let OX, OY be the original axes inclined at an angle and 0X\ OY' be the new axes inclined at an angle and let the angle XOX' = 6.
Let P be any point whose co-ordinates are x, y re- ferred to the original axes, and x\ y referred to the new axes; so that in the figure OM=x, MF=y, OM' = x, and 3rP = y', MP being parallel to OF and MT parallel
to or.
Draw PK and M'ff perpendicular to OX, and M' G perpendicular to PK.
Then KP = KG + GP = HM' + GP ;
/. 2/ sin o) = x' sin XOX' + y sin XOY
— X sin 0 + y sin (0 + o)').
Similarly, by drawing PL perpendicular to OY, we can shew that
X sin CO = X sin X'OY- y' sin YO Y'
= X sin {co — 6) — y sin {co -\- 6 — co).
These formulae are very rarely used. The results which would be obtained by the change of axes are generally found in an indirect manner, as in the following Article.
*52. If by any change of axes ax"^ + 2hxy -h by^ he changed into ax'^ + 2h'xy + b'f^, iJien will
CHANGE OF AXES. 51
a-hb — 2h cos co a ■\-h' — 2h' cos m
, ab-h' a'b'-K^
and ~^^ — = -. 2 , ,
81 ir (o sin ft)
where co and co' are the angles of inclination of the ttuo sets of axes.
If 0 be the origin and P be any point whose co-ordi- nates are a;, y referred to the old axes and x, y referred to the new, then OP^ is equal to ^^ + y^ + 2xy cos co, and also equal to x^ + 2/"* + 2xy cos o)'. Hence x^ + y + 2xy cos ca is changed into
x"^ + y"^-\- 2x'y' cos ft)'. Also, by supposition,
ax^ + 2hxy + by'^ is changed into a'x'^ + 2h'xy + b'y'^. Therefore, if \ be any constant,
ax^ + 2hxy + by^ + X (x^ + 2xy cos co + y^) will be changed into ax^ + 2Kxy' + Vy"" + \ {x" + 2xy cos ft)' + ?/''). Therefore, if X be so chosen that one of these expressions is a perfect square, the other will also be a perfect square for the same value of X.
The first will be a perfect square if
(a + X) (6 + X) - (/^ + \ cos ft))' = 0, and the second if
{a -F X) (6' -1- X) -- (// + X cos ft)')' = 0.
These two quadratic equations for finding X must have the same roots. Writing them in the forms 2 , a + b — 2h cos co ah — h^
X + . a ~ X -f- . 2 = "
sin ft) sin ft)
. .2 a +6'-2A' cosft)\ a¥-h" ,, and X' H ^-o—t X -I — . „ , = 0,
we see that
a + 6 — 27i cos co a'+b' — 2h' cos co'
—2
52 CHANGE OF AXES.
, ah — h^ a'h' — h'^ ,.,,
and — r-A - = — ^—2', ...(u).
sm « sm o)
If both sets of axes are at right angles these equations
take the simpler forms
a + h=^a+h\ and ah-h' = aV-h'' (iii).
53. The degree of an equation is not altered hy any alteration of the axes.
For, from Articles 49, 50, and 51, Ave see that, however the axes may be changed, the new equation is obtained by substituting for x and y expressions of the form
Ix' + my + 91, and I'x + m'y + n.
These expressions are of the first degree, and therefore if they replace x and y in the equation the degree of the equation will not be raised. Neither can the degree of the equation be loivered, for, if it were, by returning to the original axes, and therefore to the original equation, the degree would be raised.
Ex. 1. Prove, by actual transformation of rectangular axes, that if ax'^ + 2hxij + hy^ become a'x'^ + 2h'xy + b'y'^, then will a + b = a' + b', and h^-ab = h'^-a'b'.
Ex. 2. If the formula for transformation from one set of axes to another with the same origin be x=mx'+mj', y=in'x' + n'y'; shew that m^ + wi'2 - 1 _ mm' n^ + n'-'-l ~li^' [.r^ + y^ + 2a;?/ cos w will become a/^ + y'^ + 2x'y' cos w'. Substitute there- fore the given expressions for x and y, and equate coefficients of x'^ and y''^ to unity, and then eliminate cos w,]
Anharmonic or Cross Ratios.
*54. A set of points on a straight line is called a range ; and a set of straight lines passing through a point is called a jjencil ; each line is called a ray of the pencil.
If P, Q, R, S be four points on a straight line, the
ANHARMONIC OR CROSS RATIOS.
53
FQ PS ratio ^^ : ^^^1' FQ . ES : FS.RQ is called the anhar-
monic ratio or cross ratio of the range P, Q, R, S, and is expressed by the notation {PQRS}.
If OP, OQ, OR, 08 he a pencil of four straight lines the ratio sin POQ . sin ROS : siaPOS . sin ROQ is called the anharmonic ratio or cross ratio of the pencil, and is expressed by the notation 0{PQRS}.
If the cross ratio of a pencil or of a range is equal to — 1 it is said to be hai^monic.
It is easy to shew that if [PQRS] = — 1, then PQ:PS::PR-PQ : PS - PR, so that PQ, PR, PS are in harmonical progression.
If P, Q, R, S he Si harmonic range, then Q and >Sf are said to be harmonically conjugate with respect to P and R.
*55. If four straight lines intersecting in a point 0 he cut hy any straight line in the points P, Q, R, S, the cross ratio of the range P, Q, R, S ivill he equal to that of the pencil OP, OQ, OR, OS.
For, if ^ be the length of the perpendicular from 0 on the line PQRS, we have
64 ANHARMONIC OR CROSS RATIOS.
p .PQ = OP.OQ^mPOQ,
p.US=OR.OSsmROS,
p.PS=OP.OSsmPOS,
p.EQ==OQ.ORsmROQ.
PQ.RS^ sin POQ . sin ROS Mence ^^ ^^ ^.^ ^^^ ^ ^.^ ^^^ ;
that is [PQRS] = 0 {PQJK/Sf}.
If the pencil be cut by any two straight lines in the points P, Q, R, S and P', Q\ R', S' respectively, as in the figure, the cross ratios of the ranges P, Q, R, S and P\ Q', R\ S' will be equal to one another, since they are both equal to the cross ratio of the pencil.
If we draw the transversal P" Q" R" parallel to OS, it will meet OS at an infinitely distant point, and, represent- ing this point by the symbol oo , we have
P" O' B" oTi P"0"
0{PQRS) = iF'Q"R" CO ! = ^^^tff , = 54 ;
R"oo . smce -pr, — IS unity.
*56. To find the cross ratio of the pencil for^ned hy the lines whose equations are
x — O, y — mx = 0, 2/ = 0 and y — m'x = 0.
Draw the line x — h cutting y — mx = 0 in P, the axis of X in N, and y — mx = 0 in Q.
Then NQ = m'h, and NP = mh.
O{rPX0 = {ooPi^Q}=^^^« ^ = ^^J = ^'. ^ ^* ooQ.NP NP mh m
From the above we see that the four lines
x = 0, y — mx =0; y = 0 and y + mx = 0
form a harmonic pencil.
If the axes are at right angles to one another the lines y — mx = 0, and y + mx = 0 make equal angles with either axis.
Hence, if a pencil be harmonic and two alternate rays be at right angles to one another, they will bisect the internal and external angles between the other two.
ANHARMONIC OR CROSS RATIOS. 55
*57. To find the cross ratio of the pencil formed hy the four lines
y = kxy y = lx, y^ mx^ y = nx.
o H X
Draw any line parallel to OY cutting the given lines in the points K, L, M, N respectively, and the axis of x in jy, and let OH==x.
Then 0{KLMN}=^^.
Now KL = HL- UK = y - kx\
MN = HN- HM= nx-mx, KN = H]Sr- HK= nx- kx , ML=HL- HM= W - mx.
Hence 0{KLMN}=^^j^^-^^ ,
{k-n){m — I)
*58. To find the condition that the lines given by the two equations ax^ + 2hxy + hy^ =0 and a' a? + 2Kxy + h'y^ = 0 may be harmonically conjugate.
Let the pairs of lines he y = ax, y = ol'x ; and y = ^x, y = p'x.
Then, ii y = ax, y = ^x, y = ax, and y=^'x form a har- monic pencil, we must have [Art. 57] (a-ff)(a'-/3') ^ _
(a-^')(«'-/3) or 2aa' + 2^/S' = (a + a')(/S + /S).
56
ANHARMONIC OR CROSS RATIOS.
But, from the given equations we have
2h
h '
Hence the condition required is
ab' + ah = 2hh\
*59. We can shew in a similar manner that the pairs of points given by the equations
aoo'^ + 2hx + b=0, and a'x^ + 2h'x + h' = 0, are harmonically conjugate if
ab' + ab = 2hh\
*60. Each of the three diagonals of a quadrilateral is divided harmonically hy the other two diagonals.
Let the straight lines QAB, QDC, PDA and PCB be
INVOLUTION. 57
the sides of the quadrilateral. The line joining the point of intersection of two of these lines with the point of intersection of the other two is called a diagonal of the quadrilateral. There are therefore three diagonals, viz. PQ,AG, 5D in the figure. We have to prove that
[AOGR] = [BODS] = [QSPR] = - 1. • Let QO cut AD in K and BG in L. Then, from Art. 55,
[AOGR] = Q{AOGR]= [AKDP] = 0[AKDP] = [GLBP} = Q[GLBP] =[GOARY And, since [A 0 GR] = { GOAR}, AO.GRGO.AR
^^ AR.GO~ GR.AO'
.-. {A0GR]==±1.
We must take the negative sign, for two of the rays coincide if the anharmonic ratio of a pencil be equal to + 1.
This follows from Art. 55, for if -^tt^t/ = 1, then P" and R'
Jti (^
are coincident.
Hence the diagonal AG is cut harmonically.
We can prove in a similar manner that the other diagonals are divided harmonically.
Involution.
*61. Def. Let 0 be a fixed point on a given straight line, and P, P' ; Q, Q'\ R, R'; &c. pairs of points on the line such that
OP . OP'=OQ,OQ'=OR . 0R'=^ = a const. = A?.
Then these points are said to form a system in involution, of which the point 0 is called the centre. Two points such as P, P' are said to be conjugate to one another. The point conjugate to the centre is at an infinite distance.
58 INVOLUTION.
If each point be on the same side of the centre as its conjugate, there will be two points K^,K^, one on each side of the centre, sach that 0K^= OK^^^OP .OF. These points K^ , if 2 ^^® called double points or foci.
It is clear that when the two foci are given the involu- tion is completely determined.
An involution is also completely determined when two pairs of conjugate points are given.
For, let a, a and 6, b' be the distances of these points from any point in the straight line upon which they lie, and let x be the distance of the centre of the involution from that point. Then we have the relation {a — x) [a —x) = [h — x) {b' — a?), or {a + a -h — b') x=: aa' — bb'.
Hence there is only one position of the centre.
The position of the centre can be found geometrically by drawing circles one through each of the two pairs of conjugate points, then [Euclid ill. 37] the common chord of the circles will cut the line on which the points lie in the required centre.
*62. If any number of points be in involution the cross ratio of any four points is equal to that of their four con- jugates.
Let P, Q, R, S be any four points, and let the distances of these points from the centre be p, q, r, s respectively and
therefore those of their conjugates - , -, - , - respectively.
and [P'Q'R'S-] = ^l Pj^, l^ = jJLLSOdr^ . ^ ^ ^ /k k\/k_k\ (p-s)(r-q)
\s p) \q r)
Hence [PQRS] = [F Q'RS'].
The above gives us at once a means of testing whether
or not six points are in involution. For P, P' will be con-
INVOLUTION. 59
jugate points in the involution determined by A, A' and B, B, if [ABA'P] = [A'B'AF].
*63. Any two conjugate pomts of an involution and the tiuo foci form a harmonic range.
Let jSTj, K,^ be the two foci, and 0 the centre of the involution, and lei K^O = c= OK^.
Then, if P, P' be the two conjugates we have to prove that
K.F.K^P
or (c + OP) {OF -c)-^{c-^ OP') (OP -c) = 0,
or OP,OF = c\
which we know to be true.
*64. If any number of pairs of points in involution be joined to any point 0 we obtain a pencil of lines which may be said to be in involution.
Such a pencil is cut by any other transversal in pairs of points which are in involution. This follows from Articles 55 and 62.
EXAMPLES.
1. It P, Q, R, S he any four points on a straight line, then
PQ.ES+PE.SQ + PS.QR = 0.
2. Shew that
{ PQRS} - { QPSR } = { RSPQ } = {SRQP}.
3. Shew that
4. Shew that, by taking four points in different orders, six and only six different cross ratios are obtained, and that of these six three are the reciprocals of the other three.
5. If {PQRS]=-1, shew that {SRQP}= -1, {PRQS} = 2, and
{PSQR} = h
6. If {PQRS] = -1, and 0 be the middle point of PR, then
OP^ = OQ.OS.
7. If{PQES}=-l,shewthat^ + ^ = ^.andi^ + ± = A.
CHAPTER IV.
The Circle.
65. To find the equation of a circle referred to any rectangidar axes.
Let G be the centre of the circle, and F any point on its circumference. Let d, e be the co-ordinates of C; x, y the co-ordinates of P; and let a be the radius of the circle. Draw CM, PN parallel to OY, and C^ parallel to OX, as in the figure. Then
GK^ + KP^^GP\
THE CIRCLE. 61
But GK = x — d, and KP = y- e;
.-. (a;-dy + (y-ey = a' (i),
is the required equation.
If the centre of the circle be the origin, d and e will both be zero, and the equation of the circle will be
a,^^,f = a' (ii).
The equation (i) may be written
x'+y'- 2dx - 2ey + 1^- + e' - a' = 0. The equation of any circle is therefore of the form
a;'^ + / + 2^^H-2/y + c=0 (iii),
where g, f and c are constants.
Conversely the equation (iii) is the equation of a circle.
For it may be written
(^H-^y+(y+/)^=/+/^-o; and this last equation shews that the distance from any point on the locus of the equation (iii) from the point { — g, —f) is constant and equal to V^^ +/^ — c. The equation (iii) therefore represents a circle of radius Vo^ +y — c, the centre of the circle being at the point
If g ■\-f'^—c = 0 the radius of the circle is zero, and the circle is called a point-circle.
If g^ +f^ — c be negative, no real values of ^ and of ij will satisfy the equation, and the circle is called an imagi- nary circle.
66. We have seen that the general equation of a circle is
ic^ + if + 2gx+2fy-^c = 0. This equation contains three constants. If we want to find the equation of a circle which passes through three given points, or which is defined in some other manner, we assume the equation to be of the above form and deter- mine the values of the constants g, f, c for the circle in question from the given conditions.
62 THE CIRCLE.
Ft)r example — To find the equation of the circle which passes through the three points (0, 1), (1, 0) and (2, 1). Let the equation of the circle be
x^^-f-\-^gx^1fy + c = 0. Then, since (0, 1) is on the circle, the equation must be satisfied by putting a? = 0 and y = 1 ;
.-. 1 +2/+c = 0. Also, since (1, 0) is on the curve,
1 + 2^ + c = 0. And, since (2, 1) is on the curve,
4 + 1 + % + 2/+ c = 0. Whence ^ =/= — !, and c = 1.
The required equation is therefore
x'-V 2/' - 2a; - 2?/ + 1 = 0.
67. To find the equation of a circle when the axes are inclined at an angle co.
The square of the distance of the point (x, y) from the point (c?, e) will be equal to
{x- df + (:y- ef -{â– 2{x-d){y-e) cos o). [Art. 4.] Therefore the equation of the circle whose centre is at the point (c?, e), and whose radius is a, will be
{x-d)''+{y-ef+2{x-d){y-e)co^w = a^ (i),
or x^ + 2/^ -\-2xy cos ay — 2x {d + e cos co) — 2y (e + d cos «)
â– \- d^ + e"" + 2de cos CO - a^ = 0 (ii).
Any circle therefore referred to oblique axes has its equation of the form
x^ +y^-\- 2xy cos CO + 2gx-\- 2fy-{-c = 0 (iii),
where g, f, c are constants so long as we consider one particular circle, but are different for different circles.
The equation (iii) will still be true if we multiply throughout by any constant ; it then takes the form
Ax"" â– \-2A cosco xy + Ay'' â– \-2Gx-\-2Fy â– {- G = 0 (iv).
Hence the equation of a circle referred to oblique axes is
THE CIRCLE. 63
of the second degree, the coefficients of x"^ and y^ -are equal to one another, and the ratio of the coefficients of xy and x^ is 2 cos &>, where w is the angle between the axes.
We can find the centre and radius of the circle represented hy the equation x^ + i/ + 2xy(iOBu} + 2gx + 2fy + c = 0. For it will be identical with {x-df + {y-eY + 2{x-d){y-e)co&u}-a^=0y if d + ecosw=-/7, e + d cos w = -/, and d^ + e^ + 2de cos w-a'^ = c. We therefore have d sin^ w =/ cosu—g,e sin^ (^ = 9 cos w -/, and a^ sin^ w =/^ + i/^ — 2/j^ cos w - c sin'-^ u.
EXAMPLES.
1. Find the radii and the co-ordinates of the centres of the circles whose equations are (i) x'^ + y^-x-y = 0, (ii) 4a;^ + Ay^ +4:X-8y + S = 0.
Am. i. centre (^, ^), radius —j- ; ii. centre ( - ^, 1), radius — ^ .
2. Find the equation of the circle which passes through the points (0, 0), (a, 0) and (0, h). Ans. x^ + y^ - ax-by = 0.
3. Find the equation of the circle which passes through the points (a, 0), (- a, 0) and (0, b).
a^ - b'^ Ans. x^ + y^ + — - — y-a^ = 0.
4. Shew that, if the co-ordinates of the extremities of a diameter of a circle be {x', y') and {x", y") respectively, the equation of the circle will be {x - x') {x - x!') + (i/ - y') {y - y") = 0.
[The line joining any point P [x, y) on the circle to {x', y') makes with
the axis of x an angle tan~i^ — ^, , the line joining P to {x", y") makes
X— X
V - v" an angle tan~i ■-_- „ . Since these lines are at right angles, we have
X— X X — X
or {x-x'){x-x") + {y-y'){y-y")=Ol
5. Shew that if the co-ordinates of the extremities of a diameter be {x', y') and [x", y") respectively, the equation of the circle will be
(x - .x') {x - x") + {y- y') {y - y") + {(y - y') (.r - x") + {y - y") {x - x')} cos w
= 0, u being the angle between the axes.
64 THE CIRCLE.
6, If the equation x- + xy + y^ + 2x + 2y =0 represent a circle, shew
that the axes are inclined at an angle of 60", and find the centre and
radius of the circle.
2 Ans. centre (- |, - %); radius -—■.
7. Find the equation of the circle through the three points {x', y'), {x", y"), and {x'", y'").
68. Def, Let two points P, Q be taken on any curve, and let the point Q move along the curve nearer and nearer to the point P ; then the limiting position of the line PQ, when Q moves up to and ultimately coincides with P, is called the tangent to the curve at the point P.
The line through the point P perpendicular to the tangent is called the normal to the curve at the point P.
69. To find the equation of the tangent at any point of the circle whose equation is x^ -hy^ = a^.
Let X, y and x\ y" be the co-ordinates of two points on the circle.
The equation of the secant through the points {x, y) and (x"y y") is
x-x' _ y-y
X —X y —y
But, since the two points are on the circle, we have x'^ + y'^ = a\ and x"" + y"" = a' ;
, f2 "2 "2 '2 /"\
• • ^ -« =y -y (ii)-
Multiply the corresponding sides of the equations (i) and (ii), and we have
{x-x) {x' + x") = -{y-y) iy'+y") (iii).
Let {x', y") move up to and ultimately coincide with (xy y') ; then in the limit the chord becomes the tangent at (x\ y'). The equation of the tangent at {x\ y') is there- fore obtained by putting x" = x', and y" — y' in equation (iii) ; the result is
{x-x)x -H(y-y)2/' = 0,
THE CIRCLE. 65
or XX -^-yy =x â– \-y ]
.'. XX 4- yy = «^ is the required equation of the tangent at the point {x\ y).
70. To find the equation of the tangent at any point of the circle whose equation is
x^^-y"^ 2gx + 2/2/ + c = 0. The equation of the secant through the two points {x\ y'\ {x", y") will be
x-x' _y_-y'
a^-o^'-y -f ^'^•
Since the two points are on the circle, we have
a;'^ 4- y^-}-2^:c' + 2/^4-0-0, ^"2 ^ yu, ^ ^^^, _^ 2/2/', + c = 0,
... (^'_^")(^^^.^-+2^) = -(2/'-/)(2/'+/+2/)...(ii). _ Multiply the corresponding sides of the equations (i)
and (ii), and we get for the equation of the secant (^ _ ^') (^' + x" + 25r) = - (2/ - y') {y' 4- y"-A- 2/). The equation of the tangent at {x , y) will therefore
be
{^-^'){^' + g) + {y-y) (y +/) =o,
or xx' 4- yy +gx +fy = x"+ y"" + gx -^fy'.
Add gx+fy-\-c to both sides ; then, since {x, y') is on the circle, the equation of the tangent becomes
XX +yy'j^g{x + x) +f{y + y') + C = 0.
71. To find the equation of the normal at any point of a circle.
Let the equation of the circle be
x' 4- 2/' = ^'• If {x, y) be any point on the circle, the equation of the tangent at that point will be
xaf -\- yy = a^ (i).
The equation of the line through {x, y') perpendicular to (i) is [Art. 30]
{x-x)y -{y-y')x =0,
s.c.s. 5
66 THE CIRCLE.
or xy'—yx'=: 0 (ii).
This is the required equation of the normal at {x, y').
It is clear from equation (ii) that the normal at any point of the circle passes through the origin, that is through the centre of the circle.
72. To find the points of intersection of a given straight line and a circle.
Let the equation of the circle be
x' + y' = a' (i),
and let the equation of the straight line be
y = mx -\-c (ii).
At points which are common to the straight line and the circle both these relations are satisfied. Paints on the straight line satisfy the equation y^ = (nix + c)^ and points on the circle satisfy the equation y^ — c^—x^\ hence for the common points we have
{mx + cf = a^ - a?,
or £c''(l+m') + 2mc« + c'-a'=0 (iii).
This is a quadratic equation, and every quadratic equation has two roots, real, coincident or imaginary.
Hence there are two values of x, and the two corre- sponding values of y are found from (ii). So that every straight line meets a circle in two real, coincident, or imaginary points — imaginary points being those one or both of whose co-ordinates are imaginary.
It is impossible to represent geometrically the two imaginary points of intersection of a straight line and a circle : we shall find however that imaginary points and lines have often an important significance : and it is necessary to consider them in order to enunciate our theorems in their most general forms.
The roots of the equation (iii) will be equal to one another, if
(l+77i')(c'^-a')=mV, that is, if c' = a'(l + m'^) (iv).
THE CIRCLE. 67
If the two values of x are equal to one another the two values of y must also be equal to one another from (ii).
Therefore the two points in which the circle is cut by the line will be coincident if c = aJ\-\- wl
Hence the line y=mx + aj\+m^ will touch the circle x^ -\- y'^ = a^ for all values of m.
Since either sign may be given to the radical Jl+m^, it follows that there are two tangents to a circle for every value of m, that is, there are two tangents parallel to any given straight line.
73. To find the locus of the middle points of a system of parallel chords of a circle.
Take the centre of the circle for origin, and the axis of X parallel to the chords.
Let the equation of the circle be
x' + f = a' (i);
and let the equation of any one of the parallel chords be
y-c==0 (ii).
Where (i) and (ii) meet we have a;^ + c' = a';
.'. ^ = ± Jet"" - c\
Since the two values of x are equal and opposite, it follows that the middle point of the chord has its abscissa zero, that is, the middle point of the chord is always on the axis of y. This is true for all values of c. If c > a the two values of x are both imaginary, but their sum is still zero, and therefore the middle point of the chord is still on the axis of y.
The locus of the middle points of parallel chords of a circle is therefore the straight line through the centre which is perpendicular to the chords : the locus need not however be supposed to be limited to that portion of this line which is within the circle.
74. In the preceding Articles we have assumed no geometrical properties of the circle except that the distance
5-2
68 THE CIRCLE.
from any point to the centre is constant. Some of our results may be obtained more readily by assuming the propositions proved in Euclid, Book ill. For instance, let {x, y') be any point on the circle whose equation is x^ -\-y^ = a? \ the equation of the line from {x, y) to the
X II
centre of the circle is — — ^ = 0, and the equation of a X y^ ^
perpendicular line through {x\ y) is [Art. 80]
{x - x) x' -\-{y — y') y =0 or xx' + yy' — a^ = 0. And by Euclid lil. this line is the tangent at the point.
75. Two tangents can he drawn to a circle from any point ; and these two tangents will he real if the point he outside the circle, coincident if the point he on the circle, and imaginary if the point he within the circle. Let the equation of the circle be x^ + f = a\ and let h, h be the co-ordinates of any point. Let x\ y' be the co-ordinates of any point on the circle, then the equation of the tangent at (^', 'if) will be XX + yy — a^. The tangent at (x\ y) will pass through the point (/i, k) if
hx^^-ky'^a" (i).
But {x\ y') is on the circle, therefore
af^j^y'^^a' (ii).
Equations (i) and (ii) determine the values of x and of y' for the points the tangents at which pass through the particular point (A, k). Substitute for y' in (ii) and we get
or x'"- {h^ + 1^)- 2a^hx' + a' (a'^ - F) = 0 (iii).
Equation (iii) gives the abscissae, and from (i) we get the corresponding ordinates. Since equation (iii) is a
THE CIRCLE. 69
quadratic equation, there are two points the tangents at which pass through {Ji, k).
The roots of (iii) are real, coincident, or imaginary according as
a'h;'-a'(a'-k')(h' + k') is greater than, equal to, or less than zero. That is, according as
h' + k'-a' is greater than, equal to, or less than zero. That is, according as {h, k) is outside the circle, on the circle, or within the circle.
EXAMPLES.
1. Find the co-ordinates of the points where the line y = 2x + l cuts the circle x^ + y^ = 2. Ans. (- 1, - 1) and (^, |).
2. Shew that the line Bx-2y = 0 touches the circle x^ + y^- Sx + 2y = 0.
3. Shew that the circles x^ + y^ = 2 and x^ + y'^-6x-6y + 10 = 0 touch one another at the point (1, 1).
4. Shew that the circle x^ + y^- 2ax - 2ay + a? = 0 touches the axes of X and y.
5. Find the equation of the circle which touches the lines x = 0, y — 0, anda5=c. Ans. 4a;^ + 4?/2-4:caT±4c?/ + c^ = 0.
6. Find the equation of the circle which touches the lines x = 0,x = a, and 3a; + 41/ + 5a = 0.
Ans. x^ + y^ - ax + 2ay + a^ = 0 or x^ + y^-ax + ^ay + ^a'^ = 0.
7. Shew that the line y=m {x-a) + as/ 1 + vi^ touches the circle x^ + y^ = 2ax, whatever the value of m may be.
8. Two lines are drawn through the points (a, 0), (- a, 0) respectively, and make an angle 6 with one another ; find the locus of their intersection.
The circles x^ + y^-a^=ziz 2ay cot 6.
9. A circle touches one given straight line and cuts off a constant length {21) from another straight line perpendicular to the former ; find the equation of the locus of its centre. Ans. y'^~-x'^= 1".
10. A line moves so that the sum of the perpendiculars drawn to it from the points (a, 0), (- a, 0) is constant ; shew that it always touches a circle.
70 THE CIRCLE.
11. Find the equations of the two tangents to a^ + y^ = S, which make an angle of 60" with the axis of x. Ans. y = \/S {x±2).
12. Find the equation of the circle inscribed in the triangle the equations of whose sides are x=l, 2y = 5 and 3a;-4t/ = 5.
^?is. (x- 2)2 + (i/- 1)2 = 1.
13. Shew that the two circles
x^ + y^-2ax-2by-2ab = 0 and x^ + y^ + 2bx + 2ay -2ah = 0 cut one another at right angles. [This requires that the square of the distance between the centres of the circles is equal to the sum of the squares of their radii.]
14. Shew that the two circles represented by the equations
x^ + y^ + 2dx + k^ = 0, x^ + y^ + 2d'y -k^ = 0 intersect at right angles.
76. Tangents are drawn to a circle from any point; to find the equation of the straight line joining the points of contact of the tangents.
Let the co-ordinates of the point from which the tan- gents are drawn be x y . Let the co-ordinates of the two points of contact be h^ k and h', k' , and let x^ + y"^ — a^ — 0 be the equation of the circle.
The equations of the two tangents will be [Art. 69] och +yk — a^ — 0, xh' + yk' - a' = 0. Since both these tangents pass through the point (ic', y), therefore both equations are satisfied by the co- ordinates x\ y ;
.'. x'h + y'k-a^ = 0 (i),
and xK-\-y'k! — a^=0 (ii).
But the equations (i) and (ii) are the conditions that the two points Qi, k) and Qi, k') may lie on the line whose equation is
XX 4- yy — a^ = 0 (iii).
Hence (iii) is the required equation of the straight line through the two points of contact of the tangents which pass through {x' y').
If the equation of the circle be x^ + y^ + 2gx â– ir2fy + c^O, we can shew in a similar manner (by assuming the result of Art. 70) that the equation
THE CIRCLE. 71
of the line joining tlie points of contact of the tangents which pass through («', y') is
xaf +yy' + g {x + x') + f {y + y') + c = 0.
If the point (oc', y') be outside the circle the two tan- gents will be real, and the co-ordinates h, k and h\ k' will all be real. If however the point {a!, y') be within the circle the two tangents will be imaginary ; but, even in this case, the line whose equation is (iii) is a real line when x and y' are real. So that there is a real line joining the imaginary points of contact of the two imagi- nary tangents which can be drawn from a point within the circle.
Def. The straight line through the points of contact of the tangents (real or imaginary) which can be drawn from any point to a circle is called the polar of that point with respect to the circle.
The point of intersection of the tangents to a circle at the (real or imaginary) points of intersection of the circle and a straight line is called the pole of that line with re- spect to the circle.
77. Let TP, TQ be the two tangents to a circle from any point T. Let Q move up to and ultimately coincide with the point P, then T will also move up to and ultimately coincide with P, and the tangents TP, TQ will ultimately coincide with one another and with the chord PQ. That is to say, the polar of T, when T is on the circle, coincides with the tangent at that point.
This agrees with the result of Art. 76. For the equation of the polar is of the same form as the equation of the tangent, and hence the polar of a point which is on the circle is the tangent at that point.
72 THE CIRCLE.
78. If the polar of a pcdnt P pass through Q, then will the polar of Q pass through P.
Let P be the point {x\ y\ and Q be the point {x\ y"), and let the equation of the circle be o^ -\-y^ —c^ = 0.
The equations of the polars of (a?', y') and {x' , y') are
xx' -\-yy —a^ = Qi (i),
and xx" -h yy" — a^ = 0 (ii).
If Q be on the polar of P, its co-ordinates must satisfy the equation (i);
.-. x'x +yy-a'' = 0', but this is also the condition that P may be on the line (ii), that is on the polar of Q, which proves the proposition.
If Q be any point on a fixed straight line, and P be the pole of that line; then the polar of Q must pass through P, for by supposition the polar of P passes through Q.
Conversely, if through a fixed point P any straight line be drawn, and Q be the pole of that line ; then, since P is on the polar of Q, the point Q must always lie on a fixed straight line, namely on the polar of P.
79. If the polars of two points P, Q meet in R, then B is the pole of the line PQ. For B is on the polar of P, therefore, by Art. 78, the polar of B goes through P; similarly it goes through Q ; and therefore it must be the line PQ.
80. To give a geometrical constrniction for the polar of a point luith respect to a circle.
Let the equation of the circle be
let P be any point, and let the co-ordinates of P be x, y.
The equation of the polar of P with respect to the circle is
xx' -vyy - a^=0 (i).
The equation of the line joining P to 0, the centre of the circle, is ^ .,
^-^, = 0 (ii).
X y
THE CIRCLE.
73
We see from the equations (i) and (ii) that the polar of any point with respect to a circle is perpendicular to the line joining the point to the centre of the circle.
If OiVbe the perpendicular from 0 on the polar,
^" + ,
[Art. 31.]
ON,OP=a\
also
therefore
We have therefore the following construction for the polar. Join OP and let it cut the circle in A ; take N on the line OP such that OP : OA v. OA : ON, and draw through iV" a line perpendicular to OP.
Ex. 1. Write down the polars of the following points with respect to the circle whose equation is x^ + y^ = 4:,
(i) (2,3), (ii) (3,-1), (iii) (1,-1).
Ex. 2. Find the poles of the following lines with respect to the circle whose equation is x'^ + y^ = 35,
(i) ix + 6y-7 = 0, (ii) 3x-2y-5 = 0, (iii) ax + by -1 = 0.
Am. (i) (20, 30), (ii) (21, - 14), (iii) (35a, 35&). Ex. 3. Find the co-ordinates of the points where the line a; = 4 cuts the circle a;2 + ?/^=4; find the equations of the tangents at those points, and shew that they intersect in the point (1, 0).
Am. (4, ±,7-12), 4xd.J-i2y = 4:. Ex. 4. If the polar of the point x', y' with respect to the circle a;2 + 2/2 — op. touch the circle (« - of + y'^ = a^, shew that y'^ + lax' = a\
74 THE CIRCLE.
81. To find the polar equation of a circle.
Let G be the centre of the circle, and let its polar co-ordinates be /3, a, and let the radius of the circle be equal to a.
Let the polar co-ordinates of any point P on the curve be r, 6.
Then GP^ = 00^ + OP' - 20G , OP cos COR But GP= a, 00 = p, 0P= r, z XOG= a, / XOP = 6 ;
.-. a'=p''+r'-2rpcos{0'-a) (i),
which is the required equation.
If the origin be on the circumference of the circle p = a, and we have from (i)
?' = 2a cos (^ — a) (ii).
If, in addition, the initial line pass through the centre, a will be zero, and the equation will be
r = 2a cos 6 (iii).
From equation (i) we see that if r^, r^ be the two values of r corresponding to any particular value of 6, then
r.
\r^=^p-a' (iv),
so that r^ 7\ is independent of 0.
This proves that, if from a fixed point a straight line be drawn to cut a given circle, the rectangle contained by the segments is constant.
From (iv) we see that if the origin be within the circle, in which case p is less than a, r^ and r^ must have different signs, and are therefore drawn in different directions, as is geometrically obvious.
THE CIRCLE. 75
82. To find the length of the tangent drawn from a given point to a circle.
If T be the given point, and TP be one of the tangents from T to the circle whose centre is (7, then we know that the angle CPT is a right angle ;
. TF'^^cr-CP' (i).
Let the equation of the circle be
(a;-aY-{-{y-by-c'=0 (ii),
and let the co-ordinates of T be x\ y. Then GT = {x' - af + {y' - hf-
therefore from (i) we have
Tp-" = {x' - af + iy' -hf-c' (iii).
TP^ is therefore found by substituting the co-ordinates X, y' in the left-hand member of the equation (ii).
We see, therefore, that if ^ = 0 be the equation of a circle (where >S is written for shortness instead of x^-\-y'^ -\- 2gx -\- 2fy + c), and the co-ordinates of any point be sub- stituted in S, the result is equal to the square of the length of the tangent drawn from that point to the circle ; or [Euclid III. 37] to the rectangle of the segments of chords drawn through the point. If the point be within the circle the rectangle is negative, and the length of the tangent imaginary.
If the equation of the circle be
Ax' + Ay' + 2Gx +2Fy+G=0, to find the square of the length of the tangent from any point to the circle we must divide by A and then substi- tute the co-ordinates of the point from which the tangent is drawn.
83. If x' + y'-{-2gx +2fy +c =0 (i)
be the equation of one circle,
and x''-{-y' + 2gx -f 2fy + c = 0 (ii)
be the equation of another circle, the equation
x' + y'+2gx-\-2fy + c = x'-\-y' + 2g'x + 2fy-\-c...(m)
76 THE CIRCLE.
will clearly be satisfied by the co-ordinates of any point which is on (i) and also on (ii). Equation (iii) represents therefore some locus passing through the points common to the two circles.
But (iii) reduces to
2(^-5r> + 2(/-/Oy+C-c'=0 (iv),
which is of the first degree, and therefore represents a straight line.
Hence (iii), or (iv), is the equation of the straight line through the points common to the circles (i) and (ii).
Although the two circles (i) and (ii) may not cut one another in real points, the straight line given by (iii) or by (iv) is in all cases real, provided that ^,/, c, g ,/', c are real. We have here therefore the case of a real straight line which passes through the imaginary points of inter- section of two circles.
Another geometrical meaning can however be given to the equation (iii).
For if /Sf = 0 be the equation of a circle, in which the coefiicient of x^ is unity, and the co-ordinates of any point be substituted in /S*, the result is equal to the square of the tangent drawn from that point to the circle S=6. [Art. 82.]
Now M x,y \>Q the co-ordinates of any point on the line (iii) the left side of that equation is equal to the square of the tangent from (a?, y) to the circle (i), and the right side is equal to the square of the tangent from (x^ y) to the circle (ii).
Hence the tangents drawn to the two circles from any point of the line (iii) are equal to one another.
Def. The straight line through the (real or imaginary) points of intersection of two circles is called the radical axis of the two circles.
From the above we see that the radical axis of two circles may also be defined as the locus of the points from which the tangents drawn to the two circles are equal to one another.
THE CIRCLE. 77
The co-ordinates of the centres of the two circles are — g, -f and —g\ —f respectively: the equation of the line joining them, therefore is
9-9 "f-f which [Art. 30] is perpendicular to the line (iv).
Hence the radical axis of two circles is perpendicular to the line joining their centres. This is geometrically obvious when the circles cut in real points.
84. The three radical axes of three circles taken in pairs meet in a point.
If S=0, /S^' = 0, >Si " = 0 be the equations of three circles (in each of which the coefficient of of is unity), the equa- tion of the radical axis of the first and second will be
The equation of the radical axis of the second and third will be
And of the third and first will be
And it is obvious that if two of these equations be satisfied by the co-ordinates of any point, the third equation will also be satisfied by those co-ordinates.
The point of intersection of the three radical axes is called the radical centre of the three circles.
*85. To find the equation of a system of circles every pair of which has the same radical axis.
If the common radical axis be taken for the axis of y, and a line perpendicular to it for the axis of x, then all the circles cut a; = 0 in the same two points.
Hence, if x^+ y^ + 2gx + 2fy -f c = 0 be the general equation of the circles, when we put x = 0 the roots of the resulting equation y^ + 2fy + 0 = 0 must be the same for all the circles.
Therefore y and c must be the same for all the circles.
78 THE CIRCLE.
If we take as origin the point midway between the two points where a) = 0 cuts the circles, / will be zero, and the equation becomes
x'-{-y' + 2gx + c = 0 (i),
which is the required equation, c being the same for all the circles.
The radical axis cuts the circles in real points if c be negative, and in imaginary points if c be positive.
The equation (i) can be written
Hence, if g be taken equal to + Vc the circle will reduce to one of the points (+ fjc, 0).
These points are called the limiting points of the system of co-axial circles. When c is positive, that is when the circles themselves cut in imaginary points, the limiting points are real, and conversely, when the circles cut in real points the limiting points are imaginary.
* 86. TjT S = 0 and S' = 0 6e the equations of two circles, S - X S' = 0 will, for different values of \ represent all circles which pass through the points common ^o S = 0 and S' = 0.
For, if,Sf = Oand/Sf' = Obe
a;' + 2/' + 2^^ + 2/2/ + 0 = 0 (i),
x'-\-f-\-yx + 2fy-\-c'=^0 (ii),
then will /Sf - \ /S' = 0 be
a?' + 2/' + 2gx+2fy + c - \ {x^ +/ + 2gx-Â¥2fy + c]
= 0 (iii).
Now (iii) is clearly the equation of a circle, whatever A, may be.
Also, if the co-ordinates of any point satisfy both (i) and (ii), they will also satisfy (iii).
Hence S — \S' = 0 is, for any value of X, a circle passing through the points common to 8 = 0 and S' = 0.
By giving a suitable value to X the circle (iii) may be made to pass through any other point ; therefore S — \S'—0 represents all the circles through the inter- sections of ^ = 0 and S' = 0.
THE CIRCLE. 79
The geometrical meaning of the equation S — \S' — 0 should be noticed. From Art. 82 we see that any point whose co-ordinates satisfy the equation S = \8' is such that the square of the tangent from it to the circle >S' = 0 is equal to X times the square of the tangent from it to S' = 0. "We have therefore the following proposition — the locus of a point which moves so that the tangents from it to two given circles are in a constant ratio, is another circle which passes through their common points.
87. If 0, 0' be the centres of two circles whose radii are a, a' respectively, the two points which divide the line 00' internally and externally in the ratio a : a are called the centres of similitude of the two circles.
The properties of the centres of similitude are best treated geometrically.
The most important of the properties are (1) Two of the common tangents to two circles pass through each centre of similitude; (2) Any straight line through a centre of similitude of two circles is cut similarly by the two circles.
EXAMPLES.
1. Find the length of the tangent drawn from the point (2, 5) to the circle x^ + y^-2x-Sy -1 = 0.
Also the length of the tangents from (4, 1) to the circle
4ic? + V-3a;-2/-7 = 0. Ans. 3, 2 V3.
2. Find the equation of the circle through the points (3, 0), (0, 2) and (- 1, 1) ; and find the value of the constant rectangle of the segments of all chords through the origin. Ans. — .
3. Find the radical axis of the circles x'^ + y^ + 2x+3y-7=0 and x^ + y^-2x-y + l = 0, Ans. x + y-2 = 0.
4. Find the radical axis of the two circles x^ + y^ + bx + by-c=0 and
ax^ + ay^ + a^x + b^y = 0. . z ca ^
^ ^ Ans.ax — I)y + - = 0.
a-b
5. Find the radical axis and the length of the common chord of the circles x'^ + y^ + ax+by + c = 0 and x^ + y^ + bx + ay + c = 0.
Ans. a;-y = 0, {^(a + i)2-4c}i
80 THE CIRCLE.
6. Shew that the three circles
x^ + 1/ + Sx + ey + 12 = 0, x^ + y^ + 2x + 8y + 1G = 0, &ndi x^ + y^ + 12y + 24: = 0, have a common radical axis.
7. Eind the radical centre of the three circles
a;2 + ^/ + 4x + 7 = 0, 2x^- + 2y^ + 3x + 5y + 9 = 0, and x'^ + y^ + y = 0.
Ans. (- 2, - 1).
8. Find the equations of the straight Hnes which touch both the circles x^ + y^ = 4. and {x-4:)^ + y^=l. Find also the co-ordinates of the centres of similitude.
Ans. 3a:iV72/-8 = 0, anda;±V152/-8 = 0; (8, 0), (|, 0).
9. If the length of the tangent from (/, ^f) to the circle x^ + y^ = 6he twice the length of the tangent from ( /, g) to the circle x^ + y^ + Sx + Sy = 0, then will p + g'^ + 4f+4:g + 2 = 0.
10. If the length of the tangent from any point to the circle x'^ + y^ + 2x=0 be three times the length of the tangent from the same point to the circle x^ + y^ — 4: = 0, shew that the point must be on the circle 4x2 + 4^/2 - a; - 18 = 0.
11. Find the equation of the circle through the points of intersec- tion of the circles x^ + y^ + 2x + 3y-7==0 and x^ + y^ + 3x-2y-l = 0, and through the point (1, 2). Ans. x'^ + y^ + 4:X~7y + 5=0.
12. Find the equation of a circle through the points of intersection of a? + y^-4:=0 and x^ + y'^-2x-Ay + 4^=0 and touching the line x + 2y=0.
Atis. x^ + y^ - X - 2y=0.
*88. We shall conclude this chapter by the solution of some examples.
(1) To find the equation of the circle which cuts three given circles at right angles.
Let the equations of the given curoles be
x^ + y^ + 2gx + 2fy + c = 0 (i),
x^ + y^ + 2g'x + 2f'y + </ = 0 (ii),
x^ + y^ + 2g"x+2f"y + c" = 0 (iii).
If the circle whose equation is
ix^+y^+2Gx + 2Fy + C=0 (iv)
cut (i) at right angles, the square of the distance between their centres is equal to the sum of the squares of their radii. Hence we have (G-g)^ + {F-f)-^=G^ + F^-G+g^+P-c,
or 2Gg + 2Ff-C-c = 0 (v).
We also have, since the other circles are cut at right angles,
2Gg' + 2Ff-G-c'=0 (vi),
4 2i^"-(7-c"=0 (vii).
THE CIRCLE. 81
Eliminating G, F, C, from the equations (iv), (v), (vi), and (vii), we have for the required equation
|
a;2+«/2, |
X, |
y^ |
1 |
|
-c, |
g, |
L |
-1 |
|
-c', |
9', |
f, |
-1 |
|
-c", |
9", |
r, |
-1 |
(2) The polars of any fixed point with respect to a series of co-axial circles pass through another fixed point, and the polar of one of the limiting points of the system is the same for all the circles.
The system of circles is given by the equation
x^ + y^ + 2ax + c = 0 (i),
â– where c is the same for all the circles [Art. 85]. The limiting points of the system are {^\/c, 0).
Let the co-ordinates of the fixed point be (/, g), then the equation of the polar with respect to (i) will be
xf+yg+a{x+f) + c = 0 (ii).
And, whatever the value of a may be, the straight line (ii) always passes through the point given by xf+ yg +c = 0 and x +f= 0.
If /=±Vc and g = 0, equation (ii) reduces to f{x+f) + a{x+f) = 0; and therefore x +f= 0.
Hence the polar of one of the limiting points is the line through the other limiting point parallel to the radical axis.
(3) If ABC he any triangle, and A'B'C he the triangle formed by the polars of the three points A, B, C with respect to a circle, so that B'C is the polar of A, C'A' is the polar of B, and k'W is the polar of C ; then loill the three lines kk', BB', CC meet in a point.
Let the equation of the circle be
x'^ + y^=^a\.... (i),
and let the co-ordinates of the points A, B, G be x', y'; x'\ y" ; and x'", y'" respectively.
Then the equations of the three lines B'C, C'A', A'B' will be
xx' -^yy' -a'^ = 0 (ii),
xx" + yy" -0^ = 0 .(iii),
and xx"' + yy"'-a^ = 0 (iv).
A A' is a line through the intersection of (iii) and (iv), its equation is therefore [Art. 33] included in
xx" + yy" - a2 = X {xx'" + yy'" - a^). We find X by making the above line pass through A, whose co-ordinates are of, 7/; we get therefore
x'x" + y'y" - a2 = X [x'x'" + y'y"' - «')•
s. c. s. 6
82 THE CIRCLE.
Hence the equation of A A' is {xx" + yy" - a^){x"'x' + y"'y' - a^) - {xx'" + yy'" - a'^){x'x" + y'y" - a^) = 0. . . (v).
The other equations can now be written down from symmetry.
They will he {xx"' + yy"' - a?){x'x" + yY - a^) - {^x' + yy' - a^){x"x"' + fy"' - a2) = 0...(vi), and {xx' + yy' - aP') {x"x"' + y"y"' - a^) - {xx" + yy" - a^){x"'x' + y"'y' -a^) = 0... (vii).
Since the three equations (v), (vi), (vii) when added together vanish identically, the three lines A A', BB', GC represented by those equations must meet in a point. [Art. 34.]
(4) 0 is one of the points of intersection of two given circles, and any line through 0 cuts the circles again in the points P, Q respectively. Find^ the locus of the middle point of PQ.
Let 0 be taken for origin, and let the equations of the circles be [Art. 81]
r = 2acos (^-a), and r = 26cos(^-/3). Then, for any particular value of 6,
OP = 2a cos {d- a), and OQ = 2bcos{d-p).
If R be the middle point of PQ,
OR = i{OP+OQ); .-. OE = acos(^-a) + &co8 (^-/3). The locus of R is therefore given by
r=a cos (^-a) + &cos {d-fi) = (a cos a + 6 cos /3) cos ^ + (a sin a + 6 sin /3) sin 6. The locus is therefore the circle whose equation is r=^cos(^-JB), where A and B are given by the equations
A cos B = a cos a + & cos fi, and ^ sin J5 = a sin a + 6 sin /S.
(5) Jf from any point O on the circle circumscribing a triangle ABC, perpendiculars he drawn on the sides of the triangle, the feet of these per- pendiculars will lie on a straight line.
Take the point 0 for origin, and the diameter through it for initial line, then the equation of the circle will be 7* = 2a cos 6.
Let the angular co-ordinates of the points A, B, C be a, fi, y respec- tively.
The line BC is the line joining (2a cos /3, j9) and (2a cos 7, y). To find the polar equation of BC take the general form p = rcos {6 -<p) [Art. 45] and substitute the co-ordinates of B and of C. We thus obtain two equations to determine p and (p. The equations will be p = 2a cos ^ cos (/3 - 0), and p = 2a cos 7 cos (7 - ^).
THE CIRCLE. 83
Hence <f>-p + y, and p = 2a cos /3 cos 7. The equation of EC is therefore
2a cos /3 cos y = r cos (^ - /3 - 7) (i).
Similarly, the equations of CA and of AB will be respectively
2a cos 7 cos a = r cos (d-y-a) (ii) ,
and 2a cos aco8/3 = r cos {6 -a-^) (iii).
The co-ordinates of the feet of the perpendiculars on the lines (i), (ii), (iii), from the point O, are 2a cos j8 cos 7, fi+y; 2a cos 7 cos a, 7 + a; and 2a cos a cos /3, a + /3. These three points are all on the straight line whose equation is
2a cos a cos /3 cos y = r cos {0- a- ^-y) (iv) .
The line through the feet of the perpendiculars is called the pedal line of the point O with respect to the triangle.
Let D be another point on the circle, and let the angular co-ordinate of D be 5. The four points A, B, G, D can be taken in threes in four ways, and we shall have four pedal lines of 0 corresponding to the four triangles. We have found the equation of one of these pedal lines, viz. equation (iv). The equations of the others can be written down by sym- metry ; they will be
2a cos j8 cos 7 cos 5 = r cos (^-^3-7-5) (v),
2acos7cos5cosa = rcos (^-7-5- a) (vi),
and 2a cos 5 cos a cos p = rcoa{6-d-a-p) (vii).
The co-ordinates of the feet of the perpendiculars from O on the lines (iv), (v), (vi) and (vii) will be 2a cos a cos /3 cos 7, a + p + y, and similar expressions. These four points are all on the line whose equation is 2a cos a cos /3 cos 7 cos d = r cos {d-a- ^-y-d). This proposition can clearly be extended.
Examples on Chapter IV.
1. A point moves so that the square of its distance from a fixed point varies as its perpendicular distance from a fixed straight line; shew that it describes a circle.
2. A point moves so that the sum of the squares of its distances from the four sides of a square is constant ; shew- that the locus of the point is a circle.
3. The locus of a point, the sum of the squares of whose distances from n fixed points is constant, is a circle.
6—2
84 THE CIRCLE.
4. A, B are two fixed points, and P moves so that PA = n . PB ; shew that the locus of P is a circle. Shew also that, for different values of n, all the circles have a common radical axis.
5. Find the locus of a point which moves so that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other sides.
6. Prove that the equation of the circle circumscribing the triangle formed by the lines a; ^- y = 6, 2aj + v/ = 4, and x+2y = b is
7. Find the equation of the circle whose diameter is the common chord of the circles
a;' + 7/' + 2.r + 32/+ 1=0, and a;' + / + 4a; + Sy + 2 = 0.
8. Find the equation of the straight lines joining the origin to the points of intersection of the line x + 2?/ — 3 = 0, and the circle of + if — ^x—^y^ 0, and shew that the lines are at right angles to one another.
9. Any straight line is drawn from a fixed point 0 meeting a fixed straight line in P, and a point Q is taken on the line such that the rectangle OQ . OP is constant ; shew that the locus of ^ is a circle.
10. Anj straight line is drawn from a fixed point 0 meeting a fixed circle in P, and a point Q is taken on the line such that the rectangle OQ . OP is constant ; shew that the locus of ^ is a circle.
11. Shew that the radical axis of two circles bisects their four common tangents.
12. If 0 be one of the limiting points of a system of co-axial circles, shew that a common tangent to any two circles of the system will subtend a right angle at 0.
13. Prove that the equation of two given circles can always be put in the form
x^ + y^ -\-ax + h = 0, x^ -\-y^ + ax + h = Q, and that one of the circles will be within the other if aa and b are both positive.
THE CIKCLE. 85
14. The distances of two points from the centre of a circle are proportional to the distances of each from the polar of the other.
15. If a circle be described on the line joining the centres of similitude of two given circles as diameter, prove that the tangents drawn from any point on it to the two circles are in the ratio of the corresponding radii
16. Find the locus of a point which is such that tan- gents from it to two concentric circles are inversely as their radii.
17. If two points Af B are harmonic conjugates with respect to two others C, D, the circles on AB and CD as dia- meters cut orthogonally.
18. If two circles cut orthogonally, every diameter of either which meets the other is cut harmonically.
19. A point moves so that the sum of the squares of its distances from the sides of a regular polygon is constant ; shew that its locus is a circle.
20. A circle passes through a fixed point 0 and cuts two fixed straight lines through 0, which are at right angles to one another, in points P, Q, such that the line PQ always passes through a fixed point; find the equation of the locus of the centre of the. circle.
21. The polar equation of the circle on (a, a), (b, ft) as diameter is
r^-r{a cos {e-a) + b cos {0 - P)} + ah cos (a - y8) = 0.
22. Find the equation for determining the values of r at the points of intersection of the circle and the straight line whose equations are
r = 2a cos 6, and r cos (^ - /5) = p.
Deduce the value of p when the straight line becomes a tangent.
23. Find the co-ordinates of the centre of the inscribed circle of the triangle the equations of whose sides are
3a:-43/ = 0, 1x-Uy = 0, and 5x- 12^/ - 36 = 0.
86 THE CIRCLE.
24. Find the locus of a point the polars of which with respect to two given circles make a given angle with one another.
25. From any point on the radical axis of two circles tangents are drawn, and the lines joining the points of contact to the centres of the circles are produced to meet; find the equation of the locus of the point of intersection.
26. If the four points in which the two circles
of + y' + ax + by + c = 0, x" + y^ + ax + b'y + c' = 0 are intersected by the straight lines
Ax + J3y+ C=0, A'x + B'y-\-C' = 0 respectively, lie on another circle, then will a — a', h-h'y c- c' A, B, C =0.
A\ B\ C I
27. A system of circles is drawn through two fixed points, tangents are drawn to these circles parallel to a given straight line; find the equation of the locus of the points of contact.
28. If ^, -5, C be the centres of three co-axial circles, and ^' ^2' ^3 ^® *^® tangents to them from any point, prove the relation
BCt^^-vCAt^^ABt^^^.
29. If ^1, ^2) ^3 be the lengths of the tangents from any point to three given circles, whose centres are not in the same straight line, shew that any circle or any straight line can be represented by an equation of the form
At^^Bt^ + Ct^ = D.
What relation will hold between A, B^ C for straight lines ?
30. If a circle cut two of a system of co-axial circles at right angles, it will cut them all at right angles.
31. Shew that every circle which passes through two given points is cut orthogonally by each of a system of circles having a common radical axis.
32. Prove that all circles touching two fixed circles are orthogonal to one of two other fixed circles.
THE CIRCLE. 87
33. If two circles cut orthogonally, prove that an inde- finite number of pairs of points can be found on their common diameter such that either point has the same polar with respect to one circle that the other has with respect to the other. Also shew that the distance between such pairs of points subtends a right angle at one of the points of intersection of the two circles.
34. If the equations of two circles whose radii are «, a be S=0, S' ^0, then the circles
a a will intersect at right angles.
35. Find the locus of the point of intersection of two straight lines at right angles to one another, each of which touches one of the two circles
(x - ay + y^ = b\ (x-^ af + ^ = c% and prove that the bisectors of the angles between the straight lines always touch one or other of two other fixed circles.
36. Shew that the diameter of the circle which cuts at right angles the three escribed circles of the triangle ABC is
— 7 (1 +C0S^ COS^ + COS ^ COS (7 + cos (7 COS ^)2 .
sm A ^ '
37. Find the locus of the point of contact of two equal circles of constant radius c, each of which passes through one of two fixed points at a distance 2 a apart : and shew that, if a = c, the locus splits up into a circle of radius a and a curve whose equation may be put into the form {af + y^Y^a^ (x^—Zy^).
CHAPTER V.
The Parabola.
89. Definitions. A Conic Section, or Conic, is the locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line. The fixed point is called a focus, the fixed straight line is called a directrix, and the constant ratio is called the eccentricity.
It will be shewn hereafter [Art. 312] that if a right circular cone be cut by any plane, the section will be in all cases a conic as defined above. It was as sections of a cone that the properties of these curves were first investigated.
We proceed to find the equation and discuss some of the properties of the simplest of these curves, namely that in which the eccentricity is equal to unity. This curve is called a parabola.
90. To find the equation of a 'parabola.
Let >Sf be the focus, and let YY' be the directrix. Draw SO perpendicular to YY\ and let 0S=2a. Take 08 for the axis of x, and OY for the axis of y.
Let P be any point on the curve, and let the co- ordinates of F be X, y.
Draw PH, PM, perpendicular to the axes, as in the figure, and join SP.
THE PARABOLA.
89
Then, by definition, 8P=PM', therefore PM' = 8P' = PJST' + 8N' ; that is, x^ = f+ {x - 2a)', or 'if = A<a{x — a)
This is the required equation of the curve.
«•
The curve cuts the axis of a? at a point A where 2/ = 0 and from (i) when 2/ = 0, a? = a ; that is, OA = a. The point A is called the vertex of the parabola. - If we transfer the origin to A, the axes being un- changed in direction, equation (i) will become [Art. 49]
2/^ = 4fax (ii).
The focus is the point (a, 0). The directrix is the line x + a = 0. Also 8P^MP==0A-hAir = a + x.
91. Since the equation of the parabola is y^ = 4fax, and 1/^ is a positive quantity, x must always be positive.
90 THE PARABOLA.
and therefore the curve lies wholly on the positive side of the axis of y.
For any particular value of x there are clearly two values of y equal in magnitude, one being positive and the other negative. Hence all chords of the curve perpen- dicular to the axis of x are bisected by it, and the portions of the curve on the positive and on the negative sides of the axis of x are in all respects equal.
As X increases y also increases, and there is no limit to this increase of x and y, so that there is no limit to the curve on the positive side of the axis of y.
The line through the focus perpendicular to the direc- trix is called the axis of the parabola.
The chord through the focus perpendicular to the axis is called the latus-rectuni.
In the figure to Art. 90, SL = LK = OS = 2a. There- fore the whole length of the latus-rectum is 4a.
92. We have found that y^ — ^ax = 0 for all points on the parabola.
For all points within the curve y^ — 4iax is negative.
For, if Q be such a point, and "through Q a line be drawn perpendicular to the axis meeting the curve in P and the axis in Ny then Q is nearer to the axis than P and therefore NQ^ is less than NP^. But, P being on the curve, NP"" - 4a . AN= 0, and therefore NQ^ - 4a. J. zV is negative.
Similarly we may prove that for all points outside the curve 2/* — A<ax is positive.
Hence, if the equation of a parabola be 3/^ — 4aa; = 0, and we substitute the co-ordinates of any point in the left- hand member of the equation, the result will be positive if the point be outside the curve, negative if the point be within the curve, and zero if the point be upon the curve.
93. The co-ordinates of the points common to the straight line, whose equation is y = ttvx -H c, and the
THE PARABOLA. 91
parabola, whose equation is y^ = ^ax, must satisfy both equations.
Hence, at a common point, we have the relation,
[mx + cf — ^ax (i).
Therefore the abscissae of the common points are given by the equation (i), which may be written in the form mV-l- (2mc — 4a) ic + c'' = 0 (ii).
Since (ii) is a quadratic equation, we see that every straight line meets a parabola in two points, which may be real, coincident, or imaginary.
When vfh is very small, one root of the equation (ii) is very great ; when m is equal to zero, one root is infinitely great. Hence every straight line parallel to the axis of a parabola meets the curve in one point at a finite distance, and in another at an infinite distance from the vertex.
94. To find the condition that the line y = mx + c may touch the parabola y^ — 4tax = 0.
As in the preceding Article, the abscissae of the points common to the straight line and the parabola are given by the equation
{mx + cY = 4<ax, that is mV + (2mc — 4ta) x+c^ = 0.
If the line be a tangent, that is if it cut the parabola in two coincident points, the roots of the equation must be equal to one another. The condition for this is 4mV = (2mc - 4a)^
which reduces to mc = a, or c = — .
m
Hence, whatever m may be, the line
a
y = mx H
^ m
will touch the parabola y^ — 4ax = 0.
95. To find the equation of the straight line passing through two given points on a parabola, and to find the equation of the tangent at any point.
92 THE PARABOLA.
Let the equation of the parabola be 'if' = 4^ax, and let x, y', and x'\ y" be the co-ordinates of two points on it.
The equation of the line through these points is
y-y _ ^-^ /,-,
y-y X -X
But since the points are on the parabola, we have y'^ = ^ax , and y"^ = 4iax"'y
.-. y"^-y'^=^A^a{x'-x) (ii).
By multiplying the corresponding sides of the equa- tions (i) and (ii), we have
{y - y) {y" + y) = (^ - ^') 4^«.
or, since 3/'^ — ^ax' = 0,
y{y' + f)-'^<^^-yf = ^ (iii)>
which is the equation of the chord joining the two given points.
In order to find the equation of the tangent at (x\ y) we must put y" = y and x" = x in equation (iii), and we obtain
2yy' — 4fax —y'^ = 0, or, since y'^ = A^ax\
yy —2a{x-\-x) (iv).
Cor. The tangent at (0, 0) is x — 0) that is, the
tangent at the vertex is perpendicular to the axis.
96. We have found by independent methods [Articles 94 and 95] two
forms of the equation of a tangent to a parabola. Either of these could
however have been found from the other. Thus, suppose we know that
the equation of the tangent at (x', y') is
yy'=2a{x + x'),
,, 2a lax' then y = —x^ ;-.
y y'
If this be the same line as that given by y — mx + c
we must have
m =
y y
2a , 2aa:'
w = -T, and c= — 7-;
therefore rnc = a, as in Article
THE PARABOLA. 93
In the solution of questions we should take whichever form of the equation of a tangent appears the more suitable for the particular case.
Ex. 1. The ordinate of the point of intersection of two tangents to a pai-abola is tJie arithmetic mean between the ordinates of the points of contact of the tangents.
The equations of the tangents at the points {x', y') and {x'\ y") are yij:=2a{x + x'), and yij" = 2a (ic + x") .
By subtraction, we have for their common point, y{y'-y") = '^ax' -2.ax"
=^h{y''-y"'); .-. y=hi/+y")'
Ex. 2. To find the locus of the point of intersection of itco tangents to a parabola which are at right angles to one another. Let the equations of the two tangents be
y=^^^+'^ ^^^'
y = m^x + —, (ii).
Then, since they are at right angles, mwt'= -1. Hence the second equation can be written,
1 ,....
ij=r X- am. (ill) .
^ m ^ '
To find the abscissa of their common point we have only to subtract
(iii) from (i), and we get
0 = x(m+—\ +a(m+ — \ \ mj \ mj
and therefore we have a; +a=0.
The equation of the required locus is therefore a; + a = 0, and this [Art. 90] is the equation of the directrix.
97. To find the equation of the normal at any point of a parabola.
The equation of the tangent at [x, y) to the parabola if — 4:ax = 0, is [Art. 95]
yy=2a{x + x') (i).
The normal is the perpendicular line through [x, y). Therefore [Art. 30] its equation is
{y-y')2a^-y'{x-x')=Q (ii).
94
THE PARABOLA.
(iii).
The above equation may be written
y' ^ y"
If we put m = — ^ , then y = — 2am, and ^^ = — am^ ;
/iCli oft
therefore (iii) becomes
2/ = mx — 2am — am^ (iv).
This form of the equation of a normal is sometimes useful.
98. We will now prove some geometrical properties of a parabola.
Y
Let the tangent at the point P meet the directrix in R and the axis in T. Let PW, PM be the perpendiculars from P on the axis and on the directrix.
Let PG, the normal at P, meet the axis in G. Then, if x, y' be the co-ordinates of P, the equation of the tangent at P will be
yy'==2a{x + x) (i) [Art. .95].
Where this meets the axis, y = 0, and at that point, we have from (i), x + x = 0.
.-. TA = AN , (a);
.-. TS=AS+AN = SP (fi);
THE PARABOLA. 95
and since T8=SP, the angle 8TP is equal to the angle SPT; so that PT bisects the angle SPM (7).
We see also that the triangles RSP and RMP are equal in all respects.
Hence the angle RSP = the angle i2illfP= a right angle
- ^^\
Again, since M is the point (— a, y), and 81^ the point {a, 0), the equation of the line 8M is
-y ^a
This is clearly perpendicular [Art. 30] to the tangent at P which is given by the equation (i),
.'. 8M is perpendicular to PT (e).
Since PT is perpendicular to 8M and bisects the angle SPM, it will bisect SM. If then Z be the point of inter- section of SM and PT, SZ = ZM. But SA=AO. There- fore AZis parallel to OM, and is therefore the tangent at the vertex of the parabola; so that the line through the focus of a parabola perpendicular to any tangent PT meets PI' on the tangent at the vertex* (f).
We may prove the last proposition as follows.
Let the equation of any tangent to the parabola be
a ,....
y =.mx-\ — (ill).
The equation of the line through the focus {a, 0) perpen- dicular to (iii) is
X a ...
or ?/ = +— (iv).
The lines (iii) and (iv) clearly meet where a? = 0. The equation of the normal at P {x\ y) is [Art. 97]
ta {y — y') + y (x — x) — 0. At the point G we have y = 0, and therefore
-^ay'+y' {x-x)=0,
96 THE PARABOLA.
or 2a=x-x' = AG-AN=NG.
.'. NG = 2a ......(77).
EXAMPLES.
1. Find the equations of the tangents and the equations of the normals to the parabola y^ - 4aic=0 at the ends of its latus rectum.
Ans. x^y + a=0, y±x^Sa = 0.
2. Find the points where the line y = 3x-a cuts the parabola
y3_4aa;=0. Ans. (a,2a),(^, -%a\ .
3. Shew that the tangent to the parabola y^-Aax=zO at the point (a/, y') is perpendicular to the tangent at the point
f^ -4a3\
4. Shew that the line y = 2a; + - cuts y^ - iax = 0 in coincident points.
Shew that it also cuts 20x^ + 20y^=:a^ in coincident points.
5. A straight line touches both x^ + y^ = 2a^ and y^ = 8ax; shew that its equation is ?/ = db (aj+ 2a).
6. Shew that the line 7x + 6y= 13 is a tangent to the curve
y'i-7x-8y+U=0.
7. Shew that the equation x^-\-4:ax + 2ay=0 represents a parabola, whose vertex is at the point ( - 2a, 2a), whose latus rectum is 2a, and whose axis is parallel to the axis of y.
8. Shew that all parabolas whose axes are parallel to the axis of y have their equations of the form
a? + 2Ax + 2By + C=0.
9. Find the co-ordinates of the vertex and the length of the latus rectum of each of the following parabolas :
(i) y^=5x-t-10, (ii) x^~4x + 2y = 0,
(iii) {y - 2)2 = 5 (a + 4), and (iv) Sx^ + 12x-8y = 0.' Am. (i) (-2, 0), 5. (ii) (2, 2), 2, (iii) (-4, 2), 5. (iv) ( -2, -|), «.
10. Find the co-ordinates of the focus and the equation of the directrix of each of the parabolas in question 9.
Am. (i) (-1, 0), 4a;-f-13 = 0. (ii) (2, f), 2y-5=0. (iii) (-V, 2), 4.^-1-21 = 0. (iv) (-2, -I), 6y-fl3 = 0.
11. Write down the equation of the parabola whose focus is the origin and directrix the straight line 2a;- y- 1 = 0. Shew that the line 2y = 4jc-l touches the parabola.
THE PARABOLA. 97
12. If through a fixed point 0 on the axis of a parabola any chord POF' be drawn, shew that the rectangle of the ordinates of P and P' will be constant. Shew also that the product of the abscissae will be constant.
13. Find the co-ordinates of the point of intersection of the tangents
y=:mx+—, y=m'x+—,. Shew that the locus of their intersection is a
straight line whenever mm' is constant; and that, when ?m?i' + l=0, this line is the directrix.
14. Shew that, for all values of m, the line y=m{x^-a)+ -vnW touch the parabola y^ = 4a (a; + a).
15. Two lines are at right angles to one another, and one of them touches 2/2 = 4a (.r + a), and the other y^ = 'ia' {x-\-a') ; shew that the point of intersection of the Unes will be on the line x-\-a + a' = 0.
16. If perpendiculars be let fall on any tangent to a parabola from two given points on the axis equidistant from the focus, the difference of their squares will be constant.
17. Two straight lines AP, AQ are drawn through the vertex of a parabola at right angles to one another, meeting the curve in P, Q ; shew that the line PQ cuts the axis in a fixed point.
18. If the circle x'^-\-y'^ + Ax+By-\-G=Oc\xi the parabola if - 4aa; = 0 in four points, the algebraic sum of the ordinates of those points vrill be zero.
19. If the tangent to the parabola if - 4aa!;=0 meet the axis in T and the tangent at the vertex A in Y, and the rectangle TA YQ be completed ; shew that the locus of Q, is the parabola y^-\-ax=Q.
20. If P, Q, R be three points on a parabola whose ordinates are in geometrical progression, shew that the tangents at P, R will meet on the ordinate of Q.
21. Shew that the area of the triangle inscribed in the parabola 2/2-4a.r=0 is --(^i- 2/3) (2/2-^3) (^3-^1). where y^, y^, y.^ are the ordinates of the angular points.
99. Two tangents can he draimi to a parabola from any point, which will he real, coincident, or imaginary ^ ac- cording as. the point is outside, upon, or within the curve.
s. c. s. 7
98 THE PARABOLA.
The line whose equation is
a ...
11 = fiix H — (i)
"^ m ^ '
will touch the parabola y^ = 4aa;, whatever the value of m may be [Art. 94].
The line (i) will pass through the particular point (x, y), if
y = mx H — ,
that is if mV — my' + a = 0 (ii).
Equation (ii) is a quadratic equation which gives the directions of those tangents to the parabola which pass through the point {x\ y). Since a quadratic equation has two roots, two tangents will pass through any point [x , y).
The roots of (ii) are real, coincident, or imaginary, ac- cording as y"^ — 4^ax is positive, zero, or negative. That is [Art. 92] according as {x, y) is outside the parabola, upon the parabola, or within it.
100. To find the equation of the line through the points of contact of the two tangents which can he drawn to a parabola from any point.
Let Xj y be the co-ordinates of the point from which the tangents are drawn.
Let the co-ordinates of the points of contact of the tan- gents be /t, h and Ky k' respectively.
The equations of the tangents at (^, h) and {h\ k') are yk =^a{x-\-h) and yk' = 2a{x-\- K).
We know that {x , y') is on both these lines ;
.-. y'k = 2a{x' + h) (i)
and y'k'= 2a{x +h') (ii).
But the equations (i) and (ii) are the conditions that the points (^, k) and {h\ k') may lie on the straight line whose equation is
y'y = 2a {x -\- x) (iii).
THE PARABOLA. 99
Hence (iii) is the required equation of the line through the points of contact of the tangents from {x\ y').
The line joining the points of contact of the two tangents from any point P to a parabola is called i)iQ polar of P with respect to the parabola. [See Art. 76.]
101. If the polar of the point P with respect to a parabola pass through the point Q, then will the polar of Q pass through P.
Let the co-ordinates of P be a)\ y\ and the co-ordinates of Q be x'\ y" ,
The equation of the polar of P with respect to the parabola y^ — ^ax = 0 is
yy='La{x-\-x).
If this line pass through Q {x" , y'), we must have y"y' = 2a (x' + a?').
The symmetry of this result shews that it is also the condition that the polar of Q should pass through P.
It can be shewn, exactly as in Art. 79, that if the polars of two points P, Q meet in B, then R is the pole of the line PQ.
The equation of the polar of the focus (a, 0)isx+ a = 0. So that the polar of the focus is the directrix.
If Q be any point on the directrix, Q is on the polar of the focus S, therefore the polar of Q will pass through S, so that if tangents be drawn to a parabola from any point on the directrix the line joining the points of contact will pass through the focus.
102. The locus of the middle points of a system of parallel chords of a parabola is a straight line parallel to the axis of the parabola.
The equation of the straight line joining the two points (x\ y), (x", y") on the parabola y'^ — ^ax — ^ is [Art. 95, (iii)]
y{y +y")-^(^^-yy"=^^ (i)-
Now, if the line (i) make an angle 0 with the axis of the
7-2
100 THE PARABOLA.
tan 6 = ,, (ii).
But, if the co-ordmates of the middle point of the chord be (w, y), then will
2x = x + x'\ and 2y = y-\- y" .
Hence, from (ii), tan ^ = — ,
or y = 2a cot 6 (iii),
so that y is constant so long as 6 is constant.
Hence the locus of the middle points of a system of parallel chords of a parabola is a straight line parallel to the axis of the parabola.
Def. The locus of the middle points of a system of parallel chords of a conic is called a diameter, and the chords it bisects are called the ordinates of that diameter.
We have seen in Art. 93 that a diameter of a parabola only meets the curve in one point at a finite distance from the vertex. The point where a diameter cuts the curve is called the extremity of that diameter.
103. The tangent at the extremity of a diameter is parallel to the chords which are bisected by that diameter.
All the middle points of a system of parallel chords of a parabola are on a diameter. Hence, by considering the parallel tangent, that is the parallel chord which cuts the curve in coincident points, we see that the diameter of a system of parallel chords passes through the point of con- tact of the tangent which is parallel to the chords.
1 04. To find the equation of a parabola when the axes are any diameter and the tangent at the extremity of that diameter.
Let P be the extremity of the diameter, and let the tangent at P make an angle 6 with the axis. Then NP= 2a cot 6 [Art. 102 (iii)],
:. AN=^^ = acot'e. 4a
THE PARABOLA.
101
Let the co-ordinates of Q referred to the new axes be X, y respectively, and draw QM perpendicular to the axis of the parabola, cutting the diameter PV in K.
Then
MQ = NP+ KQ = '2a cot 6 + y sin e (i),
AM=AN+NM = AK+PV-\-VK
= a cot^ 0 + x + ycosd (ii).
But QM' = 4^a.AM',
therefore, from (i) and (ii),
(2a cotO+y sin 6y = 4a (a cot^ O + w+ycos 6), or y^ sin^ 6= ^ax (iii).
But ^i\^= acot' 6>; therefore 8P= a4-^i\^=-^.
sm^
Therefore, putting a for SP or . ^ ^ , the equation of the curve is y'^=^ax (iv).
105. If the equation of a parabola, referred to any diameter and the tangent at the extremity of that diameter
as axes, be y^ — 4a« = 0 ; the line y = mx + — will be a
tangent for all values of m ; the eqviation of the tangent
102 THE PARABOLA.
at any point (x, y') will be yy —2a{x-\-x) = 0 ; the
equation of the polar of {x , y) with respect to the
parabola will be yy' — 2a{x + x) — 0 ; and the locus of the
middle points of chords parallel to the line y = mx will
, 2a
be y — ~.
These propositions require no fresh investigations ; for Articles 94, 95, 100 and 102 hold good equally whether the axes are at right angles or not.
106. The equation of the normal at any point {x\ y') of the parabola y^ — 4tax = 0 is •
2/ -2/' +£(^-^0=0 (i).
If the line (i) pass through the point {h, k) we have
*-2''4(^-Q=" <")•
The equation (ii) gives the ordinates of the points the normals at which pass through the particular point (/?., k). The equation is a cubic equation, and therefore through any point three normals can be drawn to a parabola.
Since there is no term containing y^, we have, if y^, y^, 2/3 be the three roots of the equation (ii),
2/1 + 2/2 + ^3=0 (iii)-
Now, for a system of parallel chords of a parabola, the sum of the two ordinates at the ends of any chord is constant [Art. 102]. Therefore the normals at these points meet on the normal at a fixed point the ordinate of which added to the sum of their ordinates is zero.
Hence the locus of the intersection of the normals at the ends of a system of parallel chords of a parabola is a straight line which is a normal to the curve.
107. We shall conclude this Chapter by the solution of some examples.
(1) To find the locus of the point of intersection of two tangents to a parabola ichich make a given angle loith one another.
THE PARABOLA. 103
The line y=mx + - is a tangent to the parabola y^ - Aax^O, whatever
the value of m may be. [Art. 94.]
If {x, y) be supposed known, the equation will give the directions of the tangents which pass through that point.
The equation giving the directions will be m^x - niy + a = 0.
And, if the roots of this quadratic equation be wt^ and m^, then will
Wj + wtj = - and mjm^=i - ;
•••K-».)'='-^-
But, if the two tangents make an angle a with one another,
tana:
1 + nijin^
.-. tan2a=^-— ,. . {a + xy
So that the equation of the required locus is
y^ - 4tax - (x + a)^ tan=^ a = 0.
(2) To find the locus of the foot of tlie perpendicular dratvn from a fixed point to any tangent to a parabola.
Let the equation of the parabola be 2/^-4ckc = 0, and let h, k be the co-ordinates of the fixed point O.
The equation of any tangent to the parabola is
a y = 7nx + — (i).
The equation of a line through {h, k) perpendicular to (i) is
y-k= -- {x-h) (ii).
To find the locus we have to eliminate m between the equations (i) and (ii).
From (ii) we have w = — v ;
^ y-k'
therefore, by substituting in (i), we get
x-h y - k -
or y (y - k) {x-h) +x{x-h)^ + a (y ~ k)^=0 (iii).
The locus is therefore a curve of the third degree.
From (iii) we see that the point 0 itself is always on the locus. If
104 THE PARABOLA.
the point 0 be outside tlie parabola this presents no difficulty, for two real tangents can in that case be drawn through O, and the feet of the perpen- diculars from 0 on these will be 0 itself. When the point O is within the parabola the tangents from 0 are imaginary, and the perpendiculars to them from O are also imaginary, but they all pass through the real point 0, and therefore O is a point on the locus.
When h=a, h=0, that is when O is the focus of the parabola, (iii) reduces to x {y'^+{x-a)^}=0', so that the cubic reduces to the point circle y^+{x- a)^ = 0, and the straight line aj = 0. [See Art. 98 f].
(3) The orthocentre of the triangle formed by three tangents to a parabola is on the directrix.
Let the equations of the sides of the triangle be
y = mx+—,, y = m x + —,,. and y = m"x + —^,.
The point of intersection of the second and third sides is
/a a a \
\m"mr" ^''^^"J' The line through this point perpendicular to the first side has for equation
^~^'~^'~~^\^~ m"m"') ' Now this line outs the directrix, whose equation is x= -a, in the point whose ordinate is equal to
/111 1 \
\m m m mm m J The symmetry of this result shews that the other perpendiculars cut the directrix in the same point, which proves the theorem.
(4) 2'o find tJie locus of tlie 'point of intersection of tiro normals which are at right angles to one another.
The line whose equation is
y = mx - 2am - am^ (i)
is a normal to the parabola y^ - 4rta; = 0, whatever the value of m may be.
If the point {x, y) be supposed known, the equation (i) gives the
directions of the normals which pass through that point.
If the roots of the equation (i) be Wj, m^, W3, we have
Wj+7n2 + ^'*3=0 (ii),
2a-x .....
WjWa + mgmg + mgWj = (m),
WiWaTO3=-^^ (iv).
THE PARABOLA. 105
If two of the normals, given by Wi, m^ suppose, be perpendicular to one another, we have
Wim2= - 1 (v).
The elimination of m^, m^, m.^ from the equations (ii), (iii), (iv) and (v) will give the locus required.
The result is y^=a{x- 3a).
Examples on Chapter Y.
1 . The perpendicular from a point 0 on its polar with respect to a parabola meets the polar in the point M and cuts the axis in Of the polar cuts the axis in T^ and the ordinate through 0 cuts the curve in F, F'; shew that the points T, F, M, G, F are all on a circle whose centre is S.
2. Prove that the two parabolas y^ = ax, x^ = hy will cut one another at an angle
tan ^
2(a^ + 6t)*
3. If FSQ be a focal chord of a parabola, and FA meet the directrix in My shew that MQ will be parallel to the axis of the parabola.
4. Shew that the locus of the point of intersection of two tangents to a parabola at points on the curve whose ordinates are in a constant ratio is a parabola.
5. The two tangents from a point F to the parabola 'if—^ax=^0 make angles 6^, 0^ with the axis of x] find the locus of F (i) when tan 6^ + tan 6^ is constant, and (ii) when tan^^i + taii^d^ is constant.
6. Find the equation of the locus of the point of inter- section of two tangents to a parabola which make an angle of 45" with one another.
7. Shew that if two tangents to a parabola intercept a fixed length on the tangent at the vertex, the locus of their intersection is another equal parabola.
106 THE PARABOLA. O
8. Shew that two tangents to a parabola which make equal angles respectively with the axis and directrix but are not at right angles, intersect on the latus rectum.
9. From any point on the latus rectum of a parabola perpendiculars are drawn to the tangents at its extremities ; shew that the line joining the feet of these perpendiculars touches the parabola.
10. Shew that if tangents be drawn to the parabola y^ — 4:ax = 0 from a point on the line £c + 4a = 0, their chord of contact will subtend a right angle at the vertex.
11. The perpendicular jTiV from any point T on its polar with respect to a parabola meets the axis in M; shew that if TJ^. TM is constant the locus of jT is a parabola; shew also that if the ratio TN : TM is constant the locus is a parabola.
12. Two equal parabolas have their axes parallel and a common tangent at their vertices: straight lines are drawn parallel to the direction of either axis; shew that the locus of the middle points of the parts of the lines intercepted between the curves is an equal parabola.
13. Two parabolas touch one another and have their axes parallel; shew that, if the tangents at two points of these parabolas intersect in any point on their common tangent, the line joining their points of contact will be parallel to the axis.
14. Two parabolas have the same axis; tangents are drawn from points on the first to the second ; prove that the middle points of the chords of contact with the second lie on a fixed parabola.
15. Shew that the locus of the middle point of a chord of a parabola which passes through a fixed point is a parabola.
16. The middle point of a chord PP' is on a fixed straight line perpendicular to the axis of a parabola; shew that the locus of the pole of the chord is another parabola.
17. If TPj TQ be tangents to a parabola whose vertex is A, and if the lines AP, AT, AQ, produced if necessary, cut the directrix in^, t, and q respectively; shew that pt = tq.
THE PARABOLA. 107
18. If the diameter through any point 0 of a parabola meet any chord in P, and the tangents at the ends of that chord meet the diameter in Q, Q'; shew that 0P^= OQ . OQ'.
19. The vertex of a triangle is fixed, the base is of constant length and moves along a fixed straight line; shew that the locus of the centre of its circumscribing circle is a parabola.
20. Shew that the polar of any point on the circle
with respect to the circle
x^ +y' + 2ax -Sa^^O, will touch the parabola
i/ + iax = 0.
21. PSP' is a focal chord of a parabola ; V is the middle point of PP'y and VO is perpendicular to PP' and cuts the axis in 0; prove that SO, VO are the arithmetic and geometric means between SP' and SP.
22. PSpj QSq, PSr are three focal chords of a parabola ; QP meets the diameter through jt? in A, PP meets the diameter through q in B, and PQ meets the diameter through r in G; shew that the three points A, B, G are on a straight line through S.
23. PP' is any one of a system of parallel chords of a parabola, 0 is a point on PP such that the rectangle PO . OP' is constant; shew that the locus of (9 is a parabola.
24. On the diameter through a point 0 of a parabola two points P, P' are taken so that OP . OP' is constant; prove that the four points of intersection of the tangents drawn from P, P to the parabola will lie on two fixed straight lines parallel to the tangent at 0 and equidistant from it.
25. If a quadrilateral circumscribe a parabola the line through the middle points of its diagonals will be parallel to the axis of the parabola.
26. If from any point on a focal chord of a parabola two tangents be drawn, these two tangents are equally inclined to the tanorents at the extremities of the focal chord.
108 THE PARABOLA.
27. If two tangents to a parabola make equal angles with a fixed straight line, shew that the chord of contact must pass through a fixed point.
28. Two parabolas have a common focus and their axes in opposite directions; prove that the locus of the middle points of chords of either which touch the other is another parabola.
29. Find the locus of the middle point of a chord of a parabola which subtends a right angle at the vertex.
30. The locus of the middle points of normal chords of the parabola y^ — 4aa: = 0 is ^ + —y = x — 2a.
31. FQ is a chord of a parabola normal at P, AQ is drawn from the vertex A, and through P a line is drawn parallel to AQ meeting the axis in Ji. Shew that AH is double the focal distance of F.
32. Parallel chords are drawn to a parabola; shew that the locus of the intersection of tangents at the ends of the chords is a straight line, ^Iso the locus of the intersection of the normals is a straight line, and the locus of the intersection of these two lines is a parabola.
33. If the normals at the points F, Q, E of a. parabola meet in a point; shew that the circle FQR will go through the vertex of the parabola.
34. If the normals at two points of a parabola be inclined to the axis at angles 6, <f> such that tan 6 tan ^ = 2, shew that they intersect on the parabola.
35. The locus of a point from which two normals can be drawn making complementary angles with the axis is a parabola.
36. Two of the normals drawn to a parabola from a point F make equal angles with a given straight line. Prove that the locus of i^ is a parabola.
37. The normal at a point P of a parabola meets the axis in Gj FG is produced to H^ so that GH=^\FG\ prove that the other two normals to the parabola, which pass through ZT, are at right angles to each other.
THE PARABOLA. 109
38. The normals at three points P, Q, B oi a, parabola meet in the point 0. Prove that SP + SQ + SP+ SA = 20 M, where S is the focus and OM the perpendicular from 0 on the tangent at the vertex.
39. Any three tangents to a parabola, the tangents of whose inclinations to the axis are in any given harmonical progression, will form a triangle of constant area.
40. Shew that the area of the triangle formed by three normals to a parabola will be
2
41. If a tangent to a parabola cut two given parallel straight lines in P, Q, the locus of the point of intersection of the other tangents from P, Q to the curve will be a parabola.
42. If an equilateral triangle circumscribe a parabola, shew that the lines from any vertex to the focus will pass through the point of contact of the opposite side.
43. From any point on i/=a{x + c) tangents are drawn to y^ = 4«£c; shew that the normals to this parabola at the points of contact intersect on a fixed straight line.
44. If the normals at two points on a parabola intersect on the curve, the line joining the points will pass through a fixed point on the axis.
45. If through a fixed point any chord of a parabola be drawn, and normals be drawn at the ends of the chord, shew that the locus of the point of intersection of the normals is another parabola.
46. If three normals from a point to the parabola y^ = iax cut the axis in points whose distances from the vertex are in arithmetical progression, shew that the point lies on the curve 21ay'^2{x-2aY.
47. If three of the sides of a quadrilateral inscribed in a parabola be parallel to given straight lines; shew that the fourth side will also be parallel to a fixed straight line.
48. Circles are described on any two focal chords of a parabola as diameters. Prove that their common chord passes through the vertex of the parabola.
110 THE PARABOLA.
49. Two tangents to a given parabola make angles with the axis such that the product of the tangents of their halves is constant; prove that the locus of the point of intersection of the tangents is a confocal parabola.
50. If the circle described on the chord PQ of a parabola as diameter cut the parabola again in the points R, S, then will PQ and RS intercept a constant length on the axis of the parabola.
51. If the normals at P, Q, R meet in a point 0, and PP', QQ\ RR' be lines through P, Q, R making with the axis angles equal to those made by PO, QO, RO respectively, then will PF, QQ\ RR' pass through another point 0\ and the line 00' will be perpendicular to the polar of 0'.
52. The normals to a parabola at P, Q, R meet in 0; shew that OP . OQ . OR = a . OL . OM, where OL and OM are tangents from 0 to the parabola, and 4a is the length of the latus rectum.
53. If from any point in a straight line perpendicular to the axis of a parabola normals be drawn to the curve, prove that the sum of the squares of the sides of the triangle formed by joining the feet of these normals is constant.
54. A triangle ABC is formed by three tangents to a parabola, another triangle DEF is formed by joining the points in which the chords through two points of contact meet the diameter through the third. Shew that A^ B, C are the middle point of the sides of DEF.
55. If ABC be a triangle inscribed in a parabola, and A'B'C be a triangle formed by three tangents parallel to the sides of the triangle ABC, shew that the sides of ABC will be four times the corresponding sides of A'B'C.
56. If four straight lines touch a parabola, shew that the product of the squares of the abscissae of the point of inter- section of two of these tangents and of the point of intersection of the other two is equal to the continued product of the abscissae of the four points of contact.
THE PARABOLA. Ill
57. TP^ TQ are tangents to a parabola, and p^, j?^, p^ are the lengths of the perpendiculars from JP, T, Q respectively on any other tangent to the curve; shew that p^Vz^P^-
58. OA^ OB are tangents to a parabola, and AP, BP are the corresponding normals ; shew that, if P lies on a fixed line perpendicular to the axis, 0 describes a parabola ; and find the locus of 0, if P lies on a fixed diameter.
59. PG is the normal at P to the parabola y^-A:ttx=Q, G being on the axis; GP is produced outwards to Q so that PQ = GP; shew that the locus of ^ is a parabola, and that the locus of the intersection of the tangents at P and Q is
CHAPTER VI.
The Ellipse.
Definition. An Ellipse is the locus of a point which moves so that its distance from a fixed point, called the focus, bears a constant ratio, which is less than unity, to its distance from a fixed line, called the directrix.
108. To find the equation of an ellipse.
{ Let 8 be the focus and KL the directrix. Draw /S^^ perpendicular to the directrix. Divide ZS in A so that SA : AZ — given ratio — e'.l suppose.
There will be a point A' in ZS produced such that 8A' : A'Z :: e : 1.
THE ELLIPSE. 113
Let G be the middle point o^ AA', and let AA'= 2a. Then SA = e. AZ, and BA' = e. A'Z\
.-. 8A + SA'=^e{AZ+A'Z)- :. 2AG = 2e.ZC;
,'. ZG=- (i).
e ^^
Also SA'-8A = e (A'Z - AZ) ;
or AA' -2AS = e.AA'',
.-. SG = e.AG=ae (ii).
Now let G be taken as origin, CA' as the axis of x, and a line perpendicular to GA' as the axis of y.
Let P be any point on the curve, and let its co- ordinates be X, y.
Then, in the figure,
8P^ = e'PM'; .-. 8N' + NP^ = e'ZN\ Now SN^SG^-GN=ae-Vx; .
and ZN^ZG + GN^--\-x;
e
.'. {ae+x)'' + y^ = e''(x-{-^,
or y^ + x^(l-e^)=a^{l-e^),
5 + a-(fc^ = ^ ("^)-
Putting a? = 0, we get y=±a\/(l-e^); which gives us the intercepts on the axis of y. If these lengths be called + 6, we have
Â¥ = a'(l-e') (iv),
and the equation (iii) takes the form
s. c. s. 8
'^ â– " 1 (v).
114 THE ELLIPSE.
The latus rectum is the chord through the focus parallel to the directrix. To find its length we must put x = — ae in equation (v).
Then ' 2/2 = 6^ (1 - e^) = ^ , from (iv),
SO that the leno^th of the semi-la tus rectum is — . ° a
109. In equation (v) [Art. 108] the value of y cannot be greater than h, for otherwise w^ would be negative ; and similarly x cannot be greater than a. Hence an ellipse is a curve which is limited in all directions.
If w be numerically less than a, if' will be positive ; and for any particular value of x there will be two equal and opposite values of y. The axis of x therefore divides the curve into two similar and equal parts.
So also, if y be numerically less than h, x^ will be positive, and for any particular value of y there will be two values of x which will be equal and opposite. The axis of y therefore divides the curve into two similar and equal parts. From this it follows that if on the axis of x the points S\ Z' be taken such that G8' = SG, and CZ' = ZG, the point >Sf' will also be a focus of the curve, and the line through Z' perpendicular to GZ' will be the corresponding directrix.
If (x, y') be any point on the curve, the co-ordinates
.p2 2
X, y will satisfy the equation -2+7^ — 1 = 0; and it is
clear that in that case the co-ordinates — x , — y will also satisfy the equation, so that the point (— x\ — y) will also be on the curve. But the points {x, y) and (— x ^ —y) are on a straight line through the origin and are equidistant from the origin. Hence the origin bisects every chord which passes through it, and is therefore called the centre of the curve.
The chord through the foci is called the major aocis, and the chord through the centre perpendicular to this the minor axis.
THE ELLIPSE. 115
110. To find the focal distances of any point on an ellipse.
In the figure to Art. 108, since SP = ePM, we have
SP = eZN=^e{ZG+GN) = e{^^-\-c^ = a^ex;
also S'P=e.NZ'=e(CZ'-GN) = a-exi
* .-. SP+ST = 2a.
An ellipse is sometimes defined as the locus of a point which moves so that the sum of its distances from two fixed points is constant.
To find the equation of the curve from this definition.
Let the constant sum be 2a, and the distance between the two fixed points be 2ae.
Take the middle point of the line joining the fixed points for origin, and this line and a line perpendicular to it for axes, then we have from the given condition
J{x -aef + y^+ J{x + aef+y^=2a which, when rationalized, becomes
y^ + x^{l-e^) = a^{l-e^), which is the equation previously obtained.
111. The polar equation of the ellipse referred to the centre as pole will be found by writing r cos 0 for x, and r sinO for y in the equation
y _
— + ^ = 1
berefore be r^ cos" 6 r^ sin' (9
a' ' b' The equation will therefore be
1,
a' ¥
1 cos^^ sin'^ r\
?= -oT^-r ^'>-
The equation (i) can be written in the form
?=^+S-l)^'"'^ ••••(")•
Since ^ — -^ is positive, we see from (ii) that the least
8—2
116 THE ELLIPSE.
value of -, is -^ , and that —, increases as 6 increases from
0 to -^ , the greatest value of -j being jr, . Hence the radius vector diminishes from a to 6 as ^ increases from Oto|.
112. We have found that for all points on the ellipse
^2 ~ 7.2
V 1 = 0-
We can shew in a manner similar to that adopted in Art. 92 that, if x,y be the co-ordinates of any point within
the curve, -2 + fg — 1 will be negative , and that -2 + T2 — 1
will be positive if x, y be the co-ordinates of any point outside the curve.
113. To find the points of intersection of a given
straight line and an ellipse, and to find the condition that
a given straight line may touch the ellipse.
a? iP Note. We shall henceforth always take -2 + '^ = 1 as
the equation of the ellipse, unless it is otherwise expressed.
Let the equation of the straight line be
y = mx -\-c (i) .
At points which are common to the straight line and the ellipse both these relations are satisfied. Hence at the common points we have
x^ {mx + c)^ _ - ^+ j^ -1,
or ir« (6^ + aW) + 2mca^x -^a\c^-h^) = 0 (ii).
This is a quadratic equation, and every quadratic equation has two roots, real, coincident, or imaginary.
Hence there are two values of x, and the two corre- sponding values of ^ are given by equation (i).
THE ELLIPSE. 117
The roots of the equation (ii) will be equal to one another, if
that is, if c^ = aW + ¥.
If the two values of x are equal to one another the two values of 7/ must also be equal to one another from (i) .
Therefore the two points in which the ellipse is cut by the line will be coincident if c = \/(a^m^ + b"^).
Hence the line whose equation is
y = True + fj{a^m^ i-b^) (iii)
will touch the ellipse for all values of m.
Since either sign may be given to the radical in (iii), it follows that there are two tangents to the ellipse for every value of m, that is, there are two tangents parallel to any given straight line. These two parallel tangents are equidistant from the centre of the ellipse.
114. To find the equation of the chord joining two points on the ellipse, and to find the egioation of the tangent at any point.
Let x\ y' and x\ y" be the co-ordinates of two points on the ellipse.
The equation of the secant through the points {x , y) and {x\ y") is
^-^ _y-y /-x
X —X y —y
But, since the two points are on the ellipse, we have ^ + F = 1' and^ + ^ = l;
. a^"-^"_ y"'-y' .-.
„2 L2 V^V'
Multiply the corresponding sides of the equations (i) and (ii), and we have
(x - x) jx" + x) _ (y- y) jy" + y) a'
118 THE ELLIPSE.
or
d' '^ V '~ a' "*" v •
'2 ''2
or, since, — 2 + % = 1>
a^ + j2 "â– ^â– ^ a^ "^ 6^ ....liiij,
which is the equation of the chord joining the two given points.
In order to find the equation of the tangent at {x , y) we must put x" = x\ and y" = y' in equation (iii), and we
obtain ^^' yy'
Cor. 1. The co-ordinates of the extremities of the major axis are a, 0 and — a, 0 respectively, and, from (iv), the tangents at these points are x = a and x = — a.
Hence the tangents at the extremities of the major axis are parallel to the minor axis.
Similarly the tangents at the extremities of the minor axis are parallel to the major axis.
Cor. 2. The tangent at the point {x\ y) is parallel to the tangent at the point {—x, —y), and these two points are on a straight line through the centre of the curve.
Hence the tangents at the extremities of any chord through the centre of an ellipse are parallel to one another.
115. To find the condition that the line Ix + my — 71 = 0 inay touch the ellipse.
The equation of the lines joining the origin to the points where the ellipse
x^ ^2
^+1 = 1 «'
is cut by the straight line
lx + my = n (ii),
is [Art. 38] J + |-:-('£-t!!l^y=0 (iii).
THE ELLIPSE. Ill)
If the line (ii) cut the ellipse in coincident points, the equation (iii) will represent coincident straight lines. Therefore the left-hand member of (iii) must be a perfect square : the condition for this is
/I rwi m^_ZW
U wV W nV ~ n' '
whence aH^ + 5W = n^ (iv).
Cor. The line a; cos a + 3/ sin a— ^ = 0 will touch the ellipse, if
/ = a'cos''a + 6'sin'^a (v).
116. To find the equation of the normal at any point of an ellipse.
The equation of the tangent at any point {x , y) of the ellipse is
a^^ W ~ '
The normal is the line through {x, y) perpendicular to the tangent ; its equation is therefore [Art. 30]
X — X _y — y X y
'a'
EXAMPLES.
1. Find the eccentricities, and the co-ordinates of the foci of the following ellipses :
(i) 2:r2 + 3^2 _ 1:^0, (ii) 8 (a; -1)2 + 6 (?/ + 1)2- 1 = 0.
^''^' ^') ^3' (=*^:;76'^)' ^"^ i, (1, -1±tVV6).
2. Find the lengths of the latera recta of the ellipses in question 1.
Am. I \/2 and \ V^-
3. Shew that the line y := a; + Vf touches the ellipse 2a;2 + 3f/2 = 1.
120 THE ELLIPSE.
4. Shew that the line Sy=x — 2 cuts the curve ix^-3y^-2x = 0 in two points equidistant from the axis of y.
5. Is the point (2, 1) within or without the ellipse 2ar2 + 3y2 _ 12 z= 0?
6. Find the equations of the tangents to ^ + |2 = 1 which make an angle of eo*' with the axis of x.
7. Find (i) the equations of the tangents and (ii) the equations of the normals at the ends of the latera recta of 2x^ + 3y^ = 6.
The four points are (± 1, ±| \/S).
8. Find the equations of the tangents to -2 + r2 = l which make
equal intercepts on the axes. Ans. xi^y^Ja^-\-h^=(i.
9. Shew that the equation ^ + 2y'^=Qx represents an ellipse whose eccentricity is — ^ , and shew that the origin is at an extremity of the minor axis.
10. Find the equation of the ellipse which has the point ( - 1, 1) for focus, the line 4a;- 3^=0 for directrix, and whose eccentricity is f.
Ans. 20a52 + 24a:y + 27y2 + 72(a;-y + l):^0.
11. If the normal at the end of a latus rectum of an ellipse pass through one extremity of the minor axis, shew that the eccentricity of the curve is given by the equation «"* + e^ - 1 = 0.
12. If any ordinate MP be produced to meet the tangent at the end of the latus rectum through the focus S in. Q, shew that the ordinate of Q is equal to the distance SP.
13. A straight line AB of given length has its extremities on two fixed straight lines OA, OB which are at right angles; shew that the locus of any point G on the line is an ellipse whose semi-axes are equal to CA and GB respectively.
117. Two tangents can he drawn to an ellipse from any point, which will he real, coincident, or imaginary, according as the point is outside, upon, or within the curve.
The line whose equation is
y = mx + ^(a^m^ + lf) (i)
will touch the ellipse, whatever the value of m may be [Art. 113].
THE ELLIPSE. 121
The line (i) will pass through the particular point
that is, if
{y — mxy — a^m* — 6^ = 0, or nt' {x'^ - a^) - ^mx'y' + y"^ - 6^ = 0 (ii).
Equation (ii) is a quadratic equation which gives the directions of those tangents to the ellipse which pass through the point [x\ y). Since a quadratic equation has two roots, two tangents will pass through any point {x\ y').
The roots of (ii) are real, coincident, or imaginary, according as
is negative, zero, or positive ; or according as -2- + tz ~ ^ i^
Cb 0
positive, zero, or negative. . That is, [Art. 112] according as {x\ y) is outside the ellipse, upon the ellipse, or within it.
118. To find the equation of the line through the points of contact of the two tangents which can he drawn to an ellipse from any point
Let X, y' be the co-ordinates of the point from which the tangents are drawn.
Let the co-ordinates of the points of contact of the tan- gents be h, k and h', k' respectively.
The equations of the tangents at Qi, k) and Qi , k') are xh yk _ a''^h'~^'
, xN vkf ^
and — 2 + S^ = 1-
a 0
We know that {x\ y) is on both these lines ; x'h y'k
••• -^+1^=1 w.
J x'h' y'k' ^ ,..^
and +-^ = 1 (iiV
ah ^ ^
122 THE ELLIPSE.
But (i) and (ii) shew that the points {h, k) and {h\ k') are both on the straight line whose equation is
x'x
yy _
J'-T-' (-)•
Hence (iii) is the required equation of the line through the points of contact of the tangents from (x', y).
The line joining the points of contact of the two tangents from any point P to an ellipse is called the polar of F with respect to the ellipse. [See Art. 76.]
119. If the polar of a point P with respect to an ellipse pass through the point Q, then will the polar of Q pass through P.
This may be proved exactly as in Art. 78.
1 20. To find the locus of the point of intersection of two tangents to an ellipse which are at right angles to one another.
The line whose equation is
y = mx 4- Ja^m^ + 6^^ ..(i)
will touch the ellipse, whatever the value of m may be.
If we suppose x and y to be known, the equation gives us the directions of the tangents which pass through the point {x, y).
The equation, when rationalized, becomes
w^ {x" -a") -27nxy ^^ y^ -Â¥ =0 (ii).
Let m^ and m^ be the roots of (ii) ; then, if the tangents be at right angles, m^m^ = — 1 ;
•• x^'-d'^ ' or x' + y' = a' + ¥ (iii).
The required locus is therefore a circle.
This circle is called the director circle of the ellipse.
THE ELLIPSE.
123
121. The circle described on the major axis of an ellipse as diameter is called the auxiliary circle.
|
^ |
b |
~— |
^ |
^ |
|
/ |
^-^^^^^ |
\ |
||
|
aI |
/ |
/' |
^ |
|
|
C |
;y |
|||
|
^ |
b' |
/ |
If the equation of the ellipse be
■»>
the equation of the auxiliary circle will be
1
3 "i 2
ct a
,(ii).
If therefore any ordinate NF of the ellipse be produced to meet the auxiliary circle in^, we have from (i) and (ii)
NF-"
= 1
CN"
0 a
NFJ)_ " Np a' Hence the ordinates of the ellipse and of the circle are in a constant ratio to one another.
The angle A'Cp is called the eccentric angle of the point F. The point p on the auxiliary circle is said to correspond to the point P on the ellipse.
124 THE ELLIPSE.
If the angle A' Cp be (f>, the co-ordinates of p will be a cos (f), a sin (^ ; and those of P will be a cos <j>, h sin <f>.
122. To find the -equation of the line joining two points whose eccentric angles are given.
Let (j>, <f> be the eccentric angles of the two points ; then the co-ordinates are a cos </>, h sin (^, and a cos (\>y h sin (f) respectively.
Hence the equation of the line joining them is ic — a cos 0 _ y — bsirKJ) a cos (\i — a cos (f> b sin <^ — 6 sin <^' '
or
^ ji y • ' J.
cos (f> 'f — sin 0
a '^ 0 ^
sin i (</) + </>') cos ^- (</) + <i>') '
.-. |cosi(<^ + f) + |sinj (<^+</)'j =cosi(</)-(^')---W'
which is the required equation.
To find the tangent at the point </>, we have to put 4>'= (l> in equation (i), and we obtain
-cos 6+7 sin 6 = 1 (ii).
a ^ 0 ^ ^ '
123. From equation (i) of the preceding article we see that if the sum of the eccentric angles of two points on an ellipse is constant and equal to 2a, the chord joining those points is always parallel to the line
- cos a + f sm a = 1 : a 0
that is, the chord is always parallel to the tangent at the point whose eccentric angle is a.
Conversely, if we have a system of parallel chords of an ellipse the sum of the eccentric angles of the extremities of any chord is constant.
THE ELLIPSE.
125
124. To find the equation of the normal at any point of an ellipse in terms of the eccentric angle of the point.
Let <\> be the eccentric angle of a point P on the ellipse ; the equation of the tangent at P is [Art. 122]
00 11
-cos<f> 4- rsin <f) = 1. a ^ 0 ^
The equation of the line through (a cos <^, h sin <^)
perpendicular to the tangent is [Art. 30]
ix — a cos 6) , — (y — b sin <f>) —. — , = 0,
^ ^^ cos </> ^^ sm <^
ax by 2 ,0
or r^ = a** — 61
cos <p sm (p
125. We will now prove some geometrical properties of an ellipse.
Let the tangent at P meet the axes of x and y in T, t respectively, and let the normal meet the axes in G, g. Draw SZ, S'Z', GK perpendicular to the tangent at P; draw also GE parallel to the tangent at P, meeting the normal in P, and the focal distance 8P in E.
|
/^ |
t |
|
|
/^ |
B 7^^^ |
p |
|
AS C |
^ |
I s' Ia' t^^ |
Then if x , y' be the co-ordinates of the point P, the equation of the tangent at P will be
a^ "^ 6^ ~ ^
.(i).
126
THE ELLIPSE.
Where this cuts the axis of oo, y = 0, and at that point we have from (i),
xx'
-. -1;
^^•^=1, or CF.CT^GA'' (a).
,(ii).
•• CA' Similarly NP. Ct=GB\...
The equation of the normal at P is x — x y — y
^ ~ y_
a* y
Where the normal cuts the axis of x, we have y = 0, and therefore from (ii),
x — x = 2^> or x = x (1 ~\=e^x ;
.'. GG = e\GJSr '. (7).
|
/^ |
t |
|
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^^/^*^ |
t>i\ |
|
|
s c |
l\/G ^ |
I S' A' |
"T\
Also, since 8G = SG+GG = ae + e V, and GS' = ae- eV, we have
SG^ ^ ae + eV _ a + ex' ^ ^ , GS' ~ ae— eV ~ a — ex'" S'P ^ therefore PG bisects the angle SPS' (S).
THE ELLIPSE. 127
Again, since PG' = GN"" + NP' = (GJSr - GGf + NP\
we have PG'' = y"' + x'^ (1 - e')
And PF=KG = — ;-i ^ [Art. 31] ;
vc
74 +
.-. PF,PG=h' (e).
The line whose equation is
y = mx-\- Aj(a%f -\-Â¥) (iii)
will touch the ellipse whatever the value of m may be.
Hence, if BZ, B'Z' be the perpendiculars from the foci on the line (iii), then [Art. 31] ^ „ - mae + V [c^w^ + 6^) , ^, „, mae + Jia^rn^ + h^)
^^= — -^^(1+^) — ' '*'"' ^^ = ~ V(i+™=) '
., 6'^.S'^' = '^?'L+Zr^«^6^ (j^).
1 + m ^^
Again, the equation of the line through >Sf perpendicular to (iii) is
my-\-X'\- ae = 0 (iv).
To find the locus of Z the point of intersection of (iii) and (iv), we must eliminate m from the two equations. The equations may be written in the form
y — mx — \/{a^m^ + 6^), and my + x= — ae.
Square both sides of these equations and add, we thus obtain
{x' + f) (1 + m') = aW +Â¥+ a'e' = a'^ (1 + m') ; therefore the locus of Z is the auxiliary circle whose equation is
a>' + f = a' iv).
We should have arrived at the same result if we had supposed the perpendicular to have been drawn from S\
128
THE ELLIPSE.
12 G. Let P be any point, and let QQ' be the polar of P. Let QQ' meet the axes in T, t Draw SZ, 8'Z\ CK and PO perpendicular to QQ' ', and let P 0 meet the axes in Gy g. Then, if x, y' be the co-ordinates of P, the equation of QQi will be [Art. 118]
XX
~7^ '
= 1
The equation of POG will therefore be [Art. 30] x-x' y-y'
y'
i).
X
a* V
(ii).
From (i) and (ii) we can prove, exactly as in the preceding Article,
(a) GN.GT^GA\ (/3) NP.Ci^^GW, (^) GG = i GN, and (8) KG.PG = h\
EXAMPLES.
,1. Shew that the focus of an ellipse is the pole of the corresponding directrix.
2. Shew that the equation of the locus of the foot of the perpen- dicular from the centre of an ellipse on a tangent is r^^a^co&^d + l)^ sin'-^.
THE ELLIPSE. 129
3. Shew that the sum of the reciprocals of the squares of any two diameters of an ellipse which are at right angles to one another is constant. [See Art. 111.]
4. If an equilateral triangle be inscribed in an ellipse the sum of the squares of the reciprocals of the diameters parallel to the sides will be constant.
5. An ellipse shdes between two straight lines at right angles to one another ; shew that the locus of its centre is a circle. [See Art. 120.]
6. If the points S', H' be taken on the minor axis of an ellipse such that S'C=CB' = CS, where C is the centre and S ia a. focus; shew that the sum of the squares of the perpendiculars from S' and H' on any tangent to the elUpse is constant.
7. Shew that the locus of the point of intersection of tangents to an ellipse at two points whose eccentric angles differ by a constant is an ellipse.
[If the tangents at 0 + a and (f>- a meet at {x', y') ; then — =cos <p sec a, ~—sm<f) sec a. Eliminate <p for the locus.]
8. The polar of a point P cuts the minor axis in t, and the perpen- dicular from P to its polar cuts the polar in the point 0 and the minor axis in g ; shew that the circle through the points t, 0, g will pass through the foci. [Prove that tG.Cg = SC . CS'.]
9. Prove that the line lx + my+n=0 is a normal to
a2 "^ 62 ^' p -t- ^2 ^2 •
-^ . ,, ax hy „ ,„ u ^ cos 6 msind
[Compare with - -r^=a^-(r; we have = —
"• cos 6 sin 6 ah
then eliminate 6.1
a^-L
10. The perpendicular from the focus of an ellipse whose centre is G on the tangent at any point P will meet the line CP on the directrix.
11. If Q be the point on the auxiliary circle corresponding to the point P on an ellipse, shew that the normals at P and Q meet on a fixed circle.
12. If Q be the point on the auxiliary circle corresponding to the point P on an ellipse, shew that the perpendicular distances of the foci S, H from the tangent at Q are equal to SP and HP respectively.
13. Shew that the area of a triangle inscribed in an ellipse is
^ab {sin(/3-7) +sin (7-a) + 8in (a-^)}, where a, ^, y are the eccentric angles of the angular points.
S. C. S. 9
130 THE ELLIPSE.
127. To find the locus of the middle points of a system of parallel chords of an ellipse.
The equation of the line through the two points {x, y) and [x", y") on the ellipse is [Art. 114]
x{x-^x) , y{y^y"). .^ .vui rn
^^ ^ P "^"^ a' ^ b' ^^'
If the line (i) is parallel to the line y — mx, then
h'x+x'' ....
m = 2 '~r-. — // (ii>
ay+y
Now, if X, y be the co-ordinates of the middle point of the chord joining {x, y') and {x\ y')y then 2x = x-\-x\ and 22/ = y +y"\ therefore, from (ii), we have
m = s- (ill).
ay
Hence the locus of the middle points of all chords
which are parallel to the line y = mx is the straight line
whose equation is
W ,. .
7 = 2 X (iv).
^ a^m ^
From (iv) we see that all diameters of an ellipse [Art. 102, Def^ pass through the centre.
Writing (iv) in the form y = m'x, we see that
mm = 5 (v).
a
It is clear from the symmetry of the relation (v) that all chords parallel to y = nix are bisected by y = mx.
Hence, if one diameter of an ellipse bisect chords paral- lel to a second, the second diameter will bisect all chords parallel to the first.
Def. Two diameters are said to be conjugate when each bisects chords parallel to the other.
128. The tangent at an extremity of any diameter is parallel to the chords bisected by that diameter,
THE ELLIPSE.
131
All the middle points of a system of parallel chords of an ellipse are on a diameter. Hence, by considering the parallel tangents, that is the parallel chords which cut the curve in coincident points, we see that the diameter of a system of parallel chords passes through the points of contact of the tangents which are parallel to the chords.
129. Let P, D be extremities of a pair of conjugate diameters; let the co-ordinates of P be x, y, and the co-ordinates of D be x\ y". The equations of CP and
|
CD are ^7 = -7 and ^7, = — ; y X y X ' hence from (v) Art. 127 we have ^-~ = |
|
|
. ^^'' , yy" _ 0 |
|
(i)-
|
/- |
^^ |
~^ |
P |
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d ./I |
n' ^^^ |
/ |
px\ m' |
|
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At M |
/ |
^ |
\ |
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|
I |
tC |
^ |
^ |
If (j>, <f> be the eccentric angles of P and i? respectively, then X =a cos </>, 3/" = 6 sin </>, x' = a cos </>', "}/' = h sin ^\ Substituting these values in (i) we have
cos </> cos <f) + sin <f) sin (f> —0,
or
,|,~^'=
9—2
132 THE ELLIPSE.
Hence the difference of the eccentric angles of two points which are at extremities of two conjugate diameters of an ellipse is a right angle.
If pCp\ dCd' be the diameters of the auxiliary circle corresponding to the diameters PGP\ DCD' of the ellipse, then jpCp\ dCdf will be at right angles to one another. Hence the co-ordinates of D and of U can be at once expressed in terms of those of P or of P'.
130. To shew that the sum of the squares of two con- jugate semi-diameters is constant.
Let P, D be extremities of two conjugate diameters of the ellipse.
Let the eccentric angle of P be <^, then the eccentric
angle of i) will be </> ± ^ [Art. 129].
Li
The co-ordinates of P will be a cos </>, h sin <^, and those of D will be a cos ( </> ± ^ ) , h sin ( <^ ± ^ ) . .-. OP' = a' cos' (/) + &' sin' <^,
and aZ>' = a"" cos'^^ (</> ± f ) + ^' «^i^' (*'* - l) '
.-. aP^+0i)' = «' + 6^
131. The area of the 'parallelogram which touches an ellipse at the ends of conjugate diameters is constant.
Let PGP\ DCU be the conjugate diameters. The area of the parallelogram which touches the ellipse at P, P; D,D'h4>CP.CD sin PCD, or 4> CD. CF where CFis the perpendicular from C on the tangent at P.
Now if the eccentric angle of P be </>, the eccentric
angle of D will be <^ + ^ .
.-. CD' = a' cos' (<#» ± I) + ^' sin' U ± |) , or Oi>' = a'sin'<^4-5«cos> (i).
THE ELLIPSE. 133
The equation of the tangent at P will be [Art. 122]
- cos <b +f sm 6 = 1. a ^ 0
... CF'= .^^ . ,^, [Art. 31], cos'<^ sm^cf)' ^ -"
-^2 +-p-
^^^= ...:„........ r (ii).
a^ sin^ (j) + 1/ cos""* <^ From (i) and (ii) we see that the area of the parallelo- gram is equal to 4tab.
132. If r, ?-' be the lengths of a pair of conjugate semi-diameters, and 6 be the angle between them, then
rr sin (9 = a6 [Art. 131].
Hence sin 0 is least when rr is greatest.
Now the sum of the squares of two conjugate diameters is constant ; hence the product will be greatest when the diameters are equal to one another.
Hence the acute angle between two conjugate diameters of an ellipse is least when the conjugate diameters are equal to one another,
133. Let the eccentric angles of the extremities P, D
IT .
of two conjugate diameters be </>, 0 ± ^ respectively; then
OP'' = «'cos''</>4-6'sin'^<^, and GB^ = a^ sin** cfy-^b^ cos^ </> ;
... CP' - CD' = (a' -Â¥) cos 2cl>.
Hence CP = CD when (f> is - or -r .
^4 4
The equations of the equal conjugate diameters are therefore
a ~ b ' Hence the equi-conjugate diameters of an ellipse are
134 THE ELLIPSE.
coincident in direction with the diagonals of the rectangle formed by the tangents at the ends of its axes.
134. Def. The two straight lines drawn from any point on an ellipse to the extremities of any diameter are called supplemental chords.
Any two supplemental chords of an ellipse are parallel to a pair of conjugate diameters.
Let the chords be formed by joining the point Q to the extremities P, P' of the diameter PCP\ Let F be the middle point of QP, and V the middle point of QP\ Then CV' and CV are conjugate, for each bisects a chord parallel to the other; and GV\ CV are parallel respec- tively to QPand QP'.
Hence QP and QP' are parallel to a pair of conjugate diameters.
135. We can find the position of a pair of conjugate diameters which make a given angle with one another.
For, draw any diameter POP" and on PP' describe a segment of a circle containing the given angle. If this circle cut the ellipse in a point Q, the angle PQP' is equal to the given angle, and QP, QP' are parallel to conjugate diameters [Art. 134].
The circle and ellipse will not however intersect in any real points besides the points P, P' if the given angle be less than that between the equi-conjugate diameters of the ellipse [Art. 132].
136. To fold the equation of an ellipse referred to any pair of conjugate diameters as axes.
Let the equation of the ellipse referred to its major and minor axes be
l^yl=i (i).
Since the origin is unaltered we substitute for x, y ex- pressions of the form Ix + my, Vx -\- my in order to obtain the transformed equation [Art. 51].
THE ELLIPSE. 135
The equation of the ellipse will therefore be of the form Ax''-\-2Hxy + Bif = \ (ii).
By supposition the axis of x bisects all chords parallel to the axis of y. Therefore for any particular value of x the two values of y found from (ii) must be equal and of opposite sign. Hence ff=0 ; the equation will therefore be of the form Ax^ -\- By^ = 1 (iii).
To obtain the lengths {a^ h') of the intercepts on the axes of X, y, we must put y = 0 and a; = 0 in (iii) ; we thus obtain Aa'^ = 1 = Bh'\
Hence the equation of an ellipse referred to conjugate diameters is
a 0 where a\ h' are the lengths of the semi-diameters.
137. By the preceding Article we see that when an ellipse is referred to any pair of conjugate diameters as axes of co-ordinates, its equation is of the same form as when its major and minor axes are the axes of co- ordinates.
It will be seen that Articles 113, 114, 115, 118 and 127, hold good when the axes of co-ordinates are any pair of conjugate diameters.
138. We shall conclude this chapter by the solution of some examples.
(1) To find ichen the area of a triangle inscribed in an ellipse is greatest.
Let the eccentric angles of P, Q, R, the angulat points of the triangle, be 01, 02' 03 5 1^* P^ Qj ^ ^® ^^6 three corresponding points on the auxiliary circle.
The areas of the triangles PQR, and pqr are [Art. 6]
a cos 01, 6 sin 01, 1 I , and ^ acos0i, asin0i, Ij ; a cos 02, b sin 02, 1 1 a cos 02, a sin 02, 1
a cos 0g, b sin 03, 1 1 a cos 0.3, a sin 03, 1 1
.'.A PQR : A pqr :: b : a.
136 THE ELLIPSE.
Hence the triangles PQR and pqr are to one another in the constant
ratio b : a. Therefore PQR is greatest when pqr is greatest.
Now Apg-r is greatest when it is an equilateral triangle; and in that
27r case </>i ~ 02 = 02 ~ 03 =*^3 '"*/*!= y
Hence when a triangle inscribed in an ellipse is a maximum, the
27r 47r
(2) If any pair of conjugate diameters of an ellipse cut the tangent at a point P in T, T'; shew that TP . PT' = CD\ where CD is the diameter conjugate to CP.
Take CP, CD for axes of x and y, then the equation of the ellipse will
a^ b^
The equation of the tangent at P (a, 0) will be aj=a. If y=mx, y = m'x be the equations of any pair of conjugate diameters, then
mm'= --i^[Art. 127] (i).
But PT=ma, and PT = m'a ;
.'. PT.Pr = viin'a^ (ii)
.-. 2T. Pr = J2, from (i).
(3) The line joining the extremities of any tioo diameters of an ellipse which are at right angles to one another will always touch a fixed circle.
Let CP, CQ be two diameters which are at right angles to one another, and let the equation of the line PQ be
X cos a + y sin a =p. The equation of the lines CP, CQ will be [Art. 38] x^ y"^ _ /x cos a + y sin ay
^+p-\^ p ; <''•
But, since the lines CP, CQ are at right angles to one another, the sum of the coefficients of x^ and y^ in (i) is zero [^Vrt. 36] ;
1 1 _J^
which shews that the perpendicular distance of the line PQ from the centre is constant.
Hence the line PQ always touches a fixed circle.
THE ELLIPSE. 137
(4) To find the locus of the poles of normal chords of an ellipse. The equation of the normal at any point 6 is
^._?^ = a2_fe2 (i).
cos ^ sin ^
The equation of the polar of any point {x', y') is
^-f'=^ (")•
The equations (i) and (ii) will represent the same straight line, if
(a2 - &2) 5' ^ « and (a2 - 62) V' J • ^ 'a^ cose' ^ 'b^ sm^'
or (a2-62\cos^ = ^, and (a2 _ &2\ gin ^ = - -^
' x' ^ ' y'
therefore, by squaring and adding the two last equations, we have Hence the equation of the locus is
Xhf (a2 - 62)2 ^ ^6^2 ^ J6^2.
(5) If a quadrilateral circumscribe an ellipse, the line through the middle points of its diagonals will pass through the centre of the ellipse.
Let the eccentric angles of the four points of contact of the tangents be a, j3, y, 8.
The equations of the tangents at the points a, /3 are
-cos a + "^sina — 1, and-cos jS + y sin 3 = 1. a b a ^ b ^
These meet in the point
/ cosjjaf^ ^sin|ja+^)
\ cos ^ (a - ^) ' cos i (a - ^) The tangents at y and 5 will meet in the point
/ cos^(7 + g) ^8in^(7 + 5)\ V cos i (7 -5)' cos ^(7 -5)/'
The co-ordinates of the middle point of the line joining these points of intersection are given by
_ a cos ^ (g + j3) cos ^ (7 - 8) + cos ^ (7 + 5) cos ^ (a - /3) ^~2 ' cos ^ (7 - 5) cos i (a - ^) '
_ 6 sin ^ ( a + /3) cos i (7 - 5) + sin ^ (7 + S) cos ^ (a - /3) ^"2""'^ cos ^(7 - 8)co3^{a-p) '
138 THE ELLIPSE.
Therefore the line joining the centre of the ellipse to this point makes with the major axis an angle the tangent of which is b sin ^ (a + /3) cos^ (7 - 5) + sin ^(y + S) cos ^ (a - /3) acosi(a + i3)cosi(7-5) + co8^(7 + o)cos^(a-^)'
which is equal to
6 sin {s-a) + sin (&-/3) + sin {s-y) + sin {s - d) a cos (» - a) + cos (s - /3) + cos (s - 7) + cos (s - 5) '
where 2s = a + /3 + 7 + 5.
The symmetry of the above result shews that the line joining the centre of the ellipse to one of the middle points of the diagonals of the quadrilateral will pass through the other two middle points.
Examples on Chapter YI.
1. If SP, S'P be the focal distances of a point P on an ellipse whose centre is C, and CD be the semi-diameter conju- gate to CP; shew that SP . S'P = CD'.
2. The tangent at a point P of an ellipse meets the tangent at A, one extremity of the axis AC A', in the point Y; shew that CF is parallel to A'P, C being the centre of the curve.
3. A point moves so that the sum of the squares of its distances from two intersecting straight lines is constant. Prove that its locus is an ellipse, and find the eccentricity in terms of the angle between the lines.
4. P, Q are fixed points on an ellipse and P any other point on the curve; V, V are the middle points of PE, QR^ and VG, V G' are perpendicular to PR, QR respectively and meet the axis in G, G'. Shew that GG' is constant.
5. A series of ellipses are described with a given focus and corresponding directrix; shew that the locus of the extremities of their minor axes is a parabola.
6. PNP' is a double ordinate of an ellipse, and Q is any point on the curve; shew that, if QP, QP' meet the major axis in M, M' respectively, CM. CM'^CA'.
THE ELLIPSE. 139
7. Lines are drawn througli the foci of an ellipse perpen- dicular respectively to a pair of conjugate diameters and intersect in Q; shew that the locus of ^ is a concentric ellipse.
8. The tangent at any point P of an ellipse cuts the equi-conjugate diameters in T, T'; shew that the triangles TCP^ TCP are in the ratio of CT^ : Cr\
9. If CQ be conjugate to the normal at P, then will CP be conjugate to the normal at Q.
10. If P, D be extremities of conjugate diameters of an ellipse, and PP\ DD' be chords parallel to an axis of the ellipse: shew that PD' and P'D are parallel to the equi- conjugates.
11. li P, I) are extremities of conjugate diameters, and the tangent at P cut the major axis in T, and the tangent at jD cut the minor axis in T'; shew that TT^ will be parallel to one of the equi-conjugates.
12. QQ' is any chord of an ellipse parallel to one of the equi-conjugates, and the tangents at Q, Q' meet in T; shew that the circle QTQ' passes through the centre.
13. In the ellipse prove that the normal at any point is a fourth proportional to the perpendiculars on the tangent from the centre and from the two foci.
14. Two conjugate diameters of an ellipse are drawn, and their four extremities are joined to any point on a given circle whose centre is at the centre of the ellipse; shew that the sum of the squares of the lengths of these four lines is constant.
15. PNP' is a double ordinate of an ellipse whose centre is (7, and the normal at P meets CP' in 0; shew that the locus of 0 is an ellipse.
IG. If the normal at any point P cut the major axis in G, shew that, for different positions of P, the locus of the middle point of PG will be an ellipse.
17. A^ A' are the vertices of an ellipse, and P any point on the curve; shew that, if PN be perpendicular to AP and PM perpendicular to A'P, M, iV being on the axis AA\ fhen will JfiV be equal to the latus rectum of the ellipse.
140 THE ELLIPSE.
18. Find the equation of the locus of a point from which two tangents can be drawn to an ellipse making angles 6^ , 6^, with the axis-major such that (1) tan ^^ + tan ^^ is constant, (2) cot 6^ + cot 6^ is constant, and (3) tan 6^ tan 6^ is constant.
19. The line joining two extremities of any two diameters of an ellipse is either parallel or conjugate to the line joining two extremities of their conjugate diameters.
20. If P and Z> are extremities of conjugate diameters of an ellipse, shew that the tangents at F and J) meet on the
ellipse -J + ^j = 2, and that the locus of the middle point of
21. A line is drawn parallel to the axis-minor of an ellipse midway between a focus and the corresponding directrix; prove that the product of the perpendiculars on it from the extremi- ties of any chord passing through that focus is constant.
22. If the chord joining two points whose eccentric angles are a, p cut the major axis of an ellipse at a distance d from the
centre, shew that tan ^ tan ^ = -j , where 2a is the length
of the major axis.
23. If any two chords be drawn through two points on the axis-major of an ellipse equidistant from the centre, shew that
tan - tan ^ tan ^ tan „ = 1, where a, /8, y, 8 are the eccentric
angles of the extremities of the chords.
24. If iS^, H be the foci of an ellipse and any point A be
taken on the curve and the chords ASB^ BUC, CSD, DUE... be
drawn and the eccentric angles of A, B,C, Df...he 0^, 6 , 0 , 0 , . . ,
e, 0 e 0 6 0
prove that tan -y tan -^ = cot -^ cot -^ = tan -^ tan ~= ... 'Z Z Z Z Z 'Z
25. Shew that the area of the triangle formed by the tangents at the points whose eccentric angles are a, /3, y respec- tively is ab tan |(y8 - y) tan J (y - a) tan ^ (a - j3).
26. Prove that, if tangents be drawn to an ellipse at
THE ELLIPSE. 141
points whose eccentric angles are ^j, <ji^, <fi^, tlie radius of the circle circumscribing the triangle so formed is
^"^^ sec "^^ "-^^- sec ^-^^^^ sec 4-"-^^' •
p, q, r being the length of the diameters of the ellipse parallel to the sides of the triangle, and a, b the semi-axes of the ellipse.
27. From any point P on an ellipse straight lines are drawn through the foci S, H cutting the corresponding direc- trices in Q, R respectively; shew that the locus of the point of intersection of QH and RS is an ellipse.
28. If P, p be corresponding points on an ellipse and its auxiliary circle, centre (7, and if CP be produced to meet the auxiliary circle in q; prove that the tangent at the point Q on the ellipse corresponding to q is perpendicular to (7p, and that it cuts off from Cp a length equal to CP.
29. If P, Q be the points of contact of perpendicular tan- gents to an ellipse, and p, qhe the corresponding points on the auxiliary circle; shew that Cp, Cq are conjugate diameters of the ellipse.
30. From the centre C of two concentric circles two radii CQ, Cq are drawn equally inclined to a fixed straight line, the first to the outer circle, the second to the inner : prove that the locus of the middle point P of Qq is an ellipse, that PQ is the normal at P to this ellipse, and that Qq is equal to the diameter conjugate to CP.
31. If o) is the difference of the eccentric angles of two points on the ellipse the tangents at which are at right angles, prove that ah sino) = A/>i, where X, /x are the semi -diameters parallel to the tangents at the points, and a, b are the semi-axes of the ellipse.
32. Two equal circles touch one another, find the locus of a point which moves so that the sum of the tangents from it to the two circles is constant.
33. Prove that the sum of the products of the perpen- diculars from the two extremities of each of two conjugate diameters on any tangent to an ellipse is equal to the square of the perpendicular from the centre on that tangent.
142 THE ELLIPSE.
34. ^ is a point on the nonnal at any point P of an ellipse whose centre is C such that the lines C-P, CQ make equal angles with the axis of the ellipse ; shew that PQ is proportional to the diameter conjugate to CP.
35. If a pair of tangents to a conic be at right angles to one another, the product of the perpendiculars from the centre and the intersection of the tangents on the chord of contact is constant.
36. Find the locus of the middle points of chords of an ellipse which all pass through a fixed point.
37. HP be any point on an ellipse and any chord PQ cut the diameter conjugate to CP in i?, then will PQ . PR be equal to half the square on the diameter parallel to PQ.
38. Find the locus of the middle points of all chords of an ellipse which are of constant length.
39. Tangents at right angles are drawn to an ellipse; find the locus of the middle point of the chord of contact.
40. If three of the sides of a quadrilateral inscribed in an ellipse are parallel respectively to three given straight lines, shew that the fourth side will also be parallel to a fixed straight line.
41. The area of the parallelogram formed by the tangents at the ends of any pair of diameters of an ellipse varies inversely as the area of the parallelogram formed by joining the points of contact.
42. If at the extremities P, Q of any two diameters OP, CQ of an ellipse, two tangents Pp, Qq be drawn cutting each other in T and the diameters produced in />, and q, then the areas of the triangles TQp, TPq will be equal.
43. From the point 0 two tangents OP, OQ are drawn to
the ellipse -g + fa = 1 j shew that the area of the triangle CPQ is equal to
b'h' + a'k' and the area of the quadrilateral OPCQ is equal to
{b'h' + a'k'-a'b')\ C being the centre of the ellipse, and h, k the co-ordinates of 0.
THE ELLIPSE. 143
44. TP, TQ are tangents to an ellipse whose centre is C, shew that the area of the quadrilateral CPTQ is ah tan ^{<j> — <f>); where a, b are the semi-axes of the ellipse, and <f>, (f> are the eccentric angles of F and Q.
45. POP' is a diameter of an ellipse and QCQ' is the corresponding diameter of the auxiliary circle; shew that the area of the parallelogram formed by the tangents at P, P\ Q, Q'
is , ,-—. — jr-r , where d> is the eccentric anffle of P.
(a - 6) sin 2<^ * ^ °
46. A parallelogram circumscribes a circle, and two of the angular points are on fixed straight lines parallel to one an- other and equidistant from the centre; shew that the other two are on an ellipse of which the circle is the minor auxiliary circle.
47. Two fixed conjugate diameters of an ellipse are met in the points P, Q respectively by two lines OP, OQ which pass through a fixed point 0 and are parallel to any other pair of conjugate, diameters; shew that the locus of the middle point of PQ is a straight line.
48. If from any point 0 in the plane of an ellipse the per- pendiculars OM, ON be drawn on the equal conjugate diameters, the direction OP of the diagonal of the parallelogram MONP will be perpendicular to the polar of 0.
49. Three points A, P, B are taken on an ellipse whose centre is G. Parallels to the tangents at A and B drawn through P meet CB and GA respectively in the points Q and R, Prove that QR is parallel to the tangent at P.
50. Find the locus of the point of intersection of normals at two points on an ellipse which are extremities of conjugate diameters.
51. Normals to an ellipse are drawn at the extremities of a chord parallel to one of the equi- conjugate diameters; prove that they intersect on a diameter perpendicular to the other equi-conjugate.
52. If normals be drawn at the extremities of any focal chord of an ellipse, a line through their intersection parallel to the axis-major will bisect the chord,
144 THE ELLIPSE.
53. If a length PQ be taken in the normal at any point F of an ellipse whose centre is (7, equal in length to the semi- diameter which is conjugate to CP, shew that Q lies on one or other of two circles.
54. Shew that, if </> be the angle between the tangents to
aj2 a
the ellipse —8 + ^ — 1=0 drawn from the point {x\ y'), then will tan <j> (x" + y" -a'- ¥) = 2ab V^" + |? " 1-
55. TF, TQ are the tangents drawn from an external
point (a;, y) to the ellipse — g + p — 1 = 0 ; shew that, if /S be a
ST _^ ^ SF.SQ~a''^ b'
focus, -aW~Qn=-a+ 1.2'
56. If two tangents to an ellipse from a point T intersect at an angle <^, shew that ST . JIT cos <^ = CT^ -a'- h^, where C is the centre of the ellipse and S^ H the foci.
57. If the perpendicular from the centre C of an ellipse on the tangent at any point F meet the focal distance SF^ produced if necessary, in R\ the locus of R will be a circle.
58. If two concentric ellipses be such that the foci of one lie on the other, and if <?, e' be their eccentricities, shew that
their axes are inclined at an angle cos"' ~ -, .
ee
59. Shew that the angle w^hich a diameter of an ellipse subtends at either end of the axis-major is supplementary to that which the conjugate diameter subtends at the end of the axis-minor.
60. If ^, 0' be the angles subtended by the axis major of an ellipse at the extremities of a pair of conjugate diameters, shew that cot*^ + cot^^' is constant.
61. If the distance between the foci of an ellipse subtend angles 2^, 1& at the ends of a pair of conjugate diameters, shew that tan" 0 + tan^ Q' is constant.
THE ELLIPSE. 145
62. If X, X' be the angles which any two conjugate diame- ters subtend at any fixed point on an ellipse, prove that cot'X + cot^X' is constant.
63. Shew that pairs of conjugate diameters of an ellipse are cut in involution by any straight line.
64. A triangle whose sides touch an ellipse and enclose it, is a minimum; shew that each side of the triangle touches at its middle point, and that the triangle formed by joining the points of contact is a maximum.
Q6. A, B, C, D are four fixed points on an ellipse, and P any other point on the curve; shew that the product of the perpendiculars from P on AB and CD bears a constant ratio to the product of the perpendiculars from P on BG and DA.
66. Find the locus of the point of intersection of two normals to an ellipse which are perpendicular to one another.
67. Find the equation of the locus of the point of inter- section of the tangent at one end of a focal chord of an ellipse with the normal at the other end.
68. Two straight lines are drawn parallel to the axis-major
of an ellipse at a distance . from it; prove that the part
^a^ - b^ of any tangent intercepted between them is divided by the point of contact into two parts which subtend equal angles at the centre.
69. PG is the normal to an ellipse at P, G being in the
major axis, GP is produced outwards to Q so that PQ = GP;
shew that the locus of Q is an ellipse whose eccentricity is
a^ — b^
-= — , „ , and find the equation of the locus of the intersection of
the tangents at P and Q.
S. C. & 10
CHAPTER VII. The Hyperbola.
Definition. The Hyperbola is the locus of a point which moves so that its distance from a fixed point, called the focus, bears a constant ratio, which is greater than unity, to its distance from a fixed straight line, called the directrix.
139. To find the equation of an hyperbola.
Let >S^ be the focus and ZM the directrix.
Draw BZ perpendicular to the directrix.
Divide Z8 in A so that SA : AZ= given ratio = e : 1 suppose. Then A is a point on the curve.
There will also be a point A' in /S'i^ produced such that
SA' : A'Z '.: e : 1. Let G be the middle point oi AA', and let AA' = 2a. Then 8A = e.AZ, and SA' = e.A'Z\
.'. SA+SA' = e{AZ+A'Z); .', 28G=2e.AC',
.: CS = ae (i).
Also SA' -SA=e{A 'Z- AZ) ,
or AA' = e{AA'-2AZ);
THE HYPERBOLA.
•. AG=e.ZC,
147
or
CZ
(ii).
Now let G be taken as origin, CA as the axis of x, and a line perpendicular to GA as the axis of y.
Let P be any point on the curve, and let its co- ordinates be X, y.
Now and
or or
Then, in the figure
.-. SN' + NP' = e'ZN'. SN=GN-GS = x-ae, a
ZB- = GN-GZ = x-
e'
{x-aey + f = e'(x-f\\ x^ f
(iii).
10—2
148 THE HYPERBOLA.
Since e is greater than unity c^i^—^) is negative; if we put — 6^ for a^ (1 — e^), the equation takes the form
The latus rectum is the chord through the focus parallel to the directrix. To find its length we must put x = ae in equation (iv).
Then / = y (e^ - 1) = 4\ since h' = a' (e' - 1) ; a
so that the length of the semi-latus rectum is — .
140. In equation (iv) [Art. 139] x^ cannot be less than 61,', for otherwise y^ would be negative.
Hence no part of the curve lies between x = — a and x = a.
If w be greater than a, y^ will be positive ; and for any particular value of x there will be two equal and opposite values of y. Therefore the axis of x divides the curve into two similar and equal parts.
For any value of y^ a? is positive; and for any particular value of y there will be two equal and opposite values of x. Therefore the axis of y divides the curve into two similar and equal parts. From this it follows that if on the axis of X the points >S^', Z' be taken such that C8' = >S(7, and CZ' = ZC, the point S' will also be a focus of the curve, and the line through Z' perpendicular to CZ' will be the corresponding directrix.
If {x, y) be any point on the curve, it is clear that the point {—X, —y) will also be on the curve. But the points \x\ y') and (— x, — y') are on a straight line through the origin and are equidistant from the origin. Hence the origin bisects every chord which passes through it, and is therefore called the centre of the curve.
From equation (iv) [Art. 139] it is clear that if x^ be greater than a", y"^ will be positive, and will get larger and larger as x^ becomes larger and larger, and there is no
THE HYPERBOLA. 149
limit to this increase of x and y. The curve is therefore shaped somewhat as in the figure to Art. 139, and consists of two infinite branches.
AA' is called the transverse axis of the hyperbola. The line through G perpendicular to AA' does not meet the curve in real points ; but, if B, B' be the points on this line such that BG = GB' = 6, the line BB' is called the conjugate axis.
141. To find the focal distances of any point on an hyperbola.
In the figure to Art. 139, since SF= ePM, we have
SP = eZN=e{GN-'CZ)=^e(x--] = ex-a:
also S'P = e. PM' = e{GN+Z'G) ^eU+^=ex-\-a', .'. ST-SP = 2a.
142. The polar equation of the hyperbola referred to the centre as pole will be found by writing rcosO for x, and r smd for y in the equation
a' b' ' The equation will therefore be
r'cos*^ r'sin'^^ ,
1 cos'(9 sin'^
?==~^--i>^- w-
The equation (i) can be written in the form
r^ = ;?-(a^ + 6-^)^"^^^ (">
We see from (ii) that -3 is greatest, and therefore r is
least, when 6 is zero. As 6 increases, 3 diminishes, and
is zero when sin'^^=-„ — r» ; so that for this value of 0, a'' + b
150 THE HYPERBOLA.
r is infinite. If sin^^ be greater than -= — -^ , — „ will be
^ a!^ + ¥ r^
negative, so that a radius vector which makes with the
axis an angle greater than sin"* — -^ — -^ does not meet the curve in real points. "^ l^ to)
143. Most of the results obtained in the preceding chapter hold good for the hyperbola, and in the proofs there given it is only necessary to change the sign of b'\ We shall therefore only enumerate them.
Let the equation of the hyperbola be
(i) The line y = mx + ^/(aV — If) is a tangent for all values of m [Art. 113].
(ii) The equation of the tangent at {x\ y') is
^-f- = l. [Art. 114.] (iii) The equation of the polar of {x , y) is
f-f = 1. [Art. 118.]
(iv) The equation of the normal at {x', y) is x — x y — y'
[Art. 116.1
X y
(v) The line Ix + my — n will touch the curve, if a'r'-bW = n^ [Art. 115].
(vi) The line x cos a + 2/ sin a =^ will touch the curve, if p^ = a^ cos^a — V sin^a [Art. 115].
(vii) The equation of the director-circle of the hjrper- bola is x^ + y^ = a^- h^ [Art. 120].
The director-circle is clearly imaginary when a is less than h, and reduces to a point when a = 6.
THE HYPERBOLA. 151
(viii) The geometrical propositions proved in Art. 125 are also true for the hyperbola.
(ix) The locus of the middle points of all chords of the hyperbola which are parallel to y = mx is the straight
line y — mx, where mw! = -^ [Art. 127].
144. The lines y = mx, y = m'x are conjugate if
mm = -s .
These two diameters meet the curve in points whose abscissae are given by the equations
The first equation gives real values of x if m be less than -., and the second gives real values if m' be less than
-. But, since mmf = —i, m and 7?/ cannot both be less
than - , nor both be greater.
Therefore, of two conjugate diameters of an hyperbola one meets the curve in real points, and the other in imaginary points.
The two conjugate diameters are coincident if w = + - .
145. Let P, D be extremities of a pair of conjugate diameters ; let the co-ordinates of P be x\ y , and the co-ordinates of D be x" , y" . We know from Art. 144 that if one of these two points be real the other will be imaginary.
The equations of GP and CJ) are
y X , y X ^ = — and 4; = -— . y X y X
Hence, from (ix) Art. 143, we have
152 THE HYPERBOLA.
^|-=o (i).
whence — — = ^-f^ ,
a b
or, since {x\ y') and {x'\ y") are both on the curve,
or
.-. x"^±^y J-\ (ii),
and .'. from (i) y" = ±- ^ J— 1 (iii).
From (ii) and (iii) we have
CP' + CD' = x'' + y" - ~ y'' - -^ x"" 0 a
""^ U'^ h'J W b'J
So that, as in the case of the ellipse, the sum of the squares of two conjugate diameters is constant.
146. Definition. An asymptote is a straight line which meets a curve in two points at infinity, but which is not altogether at infinity.
To find the asymptotes of an hyperbola. To find the abscissae of the points where the straight line y = mx-\-c cuts the hyperbola, we have the equation x^ (mx + c)'' _ a' F '
2/1 7w'\ 2mc & - „ ...
Both roots of the equation (i) will be infinite if the
THE HYPERBOLA. 153
coefficients oi x^ and of a? are both zero; that is, if
1 *^' A J A
-5 — tp = ^> and mc = 0.
Hence we must have c = 0, and m = + - .
~ a
The hyperbola -^ — 72 = 1
has therefore two real asymptotes whose equations are
y=±-oc\ or, expressed in one equation,
Ob
S-F = «- (")•
Draw lines through B, B' parallel to the transverse axis, and through A, A' parallel to the conjugate axis ; then we see from (ii) that the asymptotes are the diagonals of the rectangle so formed.
The ellipse has no real points at infinity, and therefore the asymptotes of an ellipse are imaginary.
147. Any straight line parallel to an asymptote will meet the curve in one point at infinity.
For, one root of the equation (i) Art. 146 will be in- finite, if the coefficient of x^ is zero. This will be the case
if m = + - . So that the line y = + -x -\-c meets the
curve in one point at infinity whatever the value of c may be.
148. The equation of the hyperbola which has BB' for its transverse axis and A A' for its conjugate axis is
^+C=i (i).
This hyperbola and the original hyperbola, whose equation is
;?-p=i (")'
are said to be cmijugate to one another.
154 THE HYPERBOLA.
We append some properties of a pair of conjugate hyperbolas.
(1) , The two hyperbolas have the same asymptotes.
(2) If two diameters be conjugate with respect to one of the hyperbolas, they will be conjugate with respect to the other.
This follows from the condition in (ix) Article 143.
(3) The equations of the hyperbolas (ii) and (i) can [Art. 142] be written in the forms
1 _ cos^ e sin^ d
1 _ cos' 0 sin^ e r'~ a' b' '
It is clear that if, for any value of 6, r^ is positive for one curve it is negative for the other.
Hence every diameter meets one curve in real points and the other in imaginary points ; moreover the lengths of semi-diameters of the two curves are, for all values of 6, connected by the relation r^* = — ?'/.
(4) If two conjugate diameters cut the curves (ii) and (i) in P and d respectively, then CP^ - Cd^ = a^- b\
Let x\ y be the co-ordinates of P, and ic\ y" the co-ordinates of d.
THE HYPERBOLA. 155
Then the equations of GP and Gd are
^-1 = 0, and ^,-^=0. X y ^ y
The condition for conjugate diameters, viz, mm = -^ ,
gives a^~ 6^""^ ^'''^'
x"x'' y'^y"^ or — 4— = 74 .
And, since {pd y y) is on (ii), and {x\ y") on (i), we x'^fy"^ ,\ y'^fx"" A
|
or |
X- y"\ a'~ ¥ ' J' 0/ |
..(iv), |
||
|
and, .'. from (iii), |
^ -A.y a -b " |
..(V). |
||
|
Hence GP'- |
-Gd'=x'' + y''- |
-x"'- |
-y"' |
|
|
= x' + y''- |
a' -b^y |
|||
|
= {a'-Â¥)\ |
fx' |
6V' |
||
|
:. GP' - |
-Cd' = a'-Â¥*. |
(5) The parallelogram formed by the tangents at P, P\ dy d! is of constant area.
The parallelogram is equal to 4(7P . Gd sin PGd, or equal to 4<Gd . GF, where GF is the perpendicular from G on the tangent at P.
* CP and Cd must not be looked upon as conjugate semi-diameters, since the points P and d are not on the same hyperbola. The line dCd' cuts the original hyperbola in two imaginary points ; and if these points be D, D', we see from (3) that CD^= - Cd\
156 THE HYPERBOLA.
Now the equation of the tangent at P is
|
XX |
yy ^ If |
1; 1 |
|
a' |
And C^=JyH^..'^ = avg + |;y
Hence Cd.CF^ah.
(6) The asymptotes bisect Pd and Pd\
If X, y be the co-ordinates of the middle point of Pd^ then
^x — X + x\ and 2y = y + y" ;
x_x+x _ 0^ ^ a
•^ ~ a therefore the middle points of Pd and of Pd' are on one or other of the lines
a ~ b' Also, since CPKd is a parallelogram GK bisects Pd or Pd\ and therefore is one of the asymptotes, so that the tangents at D, D' meet those at dy d' on the asymptotes.
(7) The equations of the polars of {x , y) with respect to the hyperbolas (ii) and (i) respectively are
^_.M^-1 and- — 4-&-1
Hence the polars of any point with respect to the two curves are parallel to one another and equidistant from the centre.
If {x\ y) be any point P on (ii), then its polar with respect to (i) is
_^,yy'_. ^(-^^) y{-y)_.
a' "^ h' ~ ' a' If
THE HYPERBOLA. 157
But the last equation is the tangent to (ii) at the point (— x\ — 2/'), which is the other extremity of the diameter through P.
Hence, if from any point on an hyperbola the tangents PQ^ PQ be drawn to the conjugate hyperbola, the line QQ' will touch the original hyperbola at the other end of the diameter through P.
149. To find the equation of an hyperbola referred to any pair of conjugate diameters as axes.
The equation of the hyperbola referred to its transverse and conjugate axes is
a:' ¥~ Since the origin is unaltered we substitute for cc, y ex- pressions of the form Ix + my, lx-\- m'y in order to obtain the transformed equation [Art. 51].
The equation of the hyperbola will therefore be of the form
Ax'-\-'lHxy + Bf = l (i).
By supposition the axis of x bisects the chords parallel to the axis of y. Therefore for any particular value of x the two values of y found from (i) must be equal and opposite. Hence H— 0 ; the equation will therefore be of the form
^«' + %'=l (ii).
Of the two semi-conjugate diameters one is real and the other imaginary. If their lengths be a! and J —lb' ; since these are the intercepts on the axes of x and y re- spectively, we obtain from (ii)
Aa''=-1 = -Bb'\ Hence the required equation is x^ V^
^^-f-^=i («')•
150. Since the equation of the curve is of the same form as before, all investigations in which it was not assumed
158
THE HYPERBOLA,
that the axes were at right angles to one another still hold good. For example (i), (ii), (iii), (v) and (ix) of Art. 143 require no change. Art. 146 will also apply without change, so that the equation of the asymptotes of the hyperbola whose equation is (ii) is
a'' h" '
151. To find the equation of an hyperbola when referred to its asymptotes as axes of co-ordinates.
Let the asymptotes be the lines CK, CK' in the
figure, and let the angle ACK' = a, so that tan a = - .
Let P be any point (x, y) of the curve, and let x , y be the co-ordinates of P when referred to CK, GK', Draw PM parallel to CK' to meet CK in M, and draw PN perpendicular to the transverse axis.
or
or
Then GM= x\ MP = y\ CN - x, NP = y. Now CN = CM cos a + MP cos a,
x = (x -^-y') cosa (i).
Also iYP= MPsin a - CM ain ol,
y = (y —x)sma (ii).
|
THE HYPERBOLA. Hence, by substituting in the equation we obtain cos' a {x + yy sin' a {y' -xj ^ |
....(iii). 1 |
|
But tan a——, therefore — ^ — = - „ = - |
159
Hence, suppressing the accents, we have from (iii)
^xy = a' + 6', which is the required equation.
The equation of the conjugate hyperbola, when referred to the asymptotes, will be
4^2/ = -(a' +5').
152. The equations of an hyperbola, of the asymptotes, and of the conjugate hyperbola are
^_^-l ^'_^-o and--^'--l
respectively.
If the axes of co-ordinates be changed in any manner, we should, in order to obtain the new equations, have to make the same substitutions in all three cases.
Hence, for all positions of the axes of co-ordinates, the equations of an hyperbola and of the conjugate hyperbola will only differ from the equation of the asymptotes by constants, and the two constants will be equal and opposite for the two hyperbolas.
158. To find the equation of the tangent at any point of the hyperbola whose equation is ^xy = a' + ¥.
The equation of the line joining the two points {x, y)y
{X, y") is
y-y' ^ ^-^^ (\\
y"-y' xr-x' â– ^^'
160 THE HYPERBOLA.
But, since the points {x, y') and {x'\ y") are on the
curve,
,, a' + W _, xy = —^—==xy ;
••• y -y = -#-^'
y X
From (i) and (ii) we have
y — y_ x — x
"7 ^-
The equation of the tangent at {x , y) is therefore
y ^
or - + ^=2 (m).
a; y
From (iii) we see that the intercepts on the a.xes are 1x and ^y.
Hence the portion of the tangent intercepted by the asymptotes is bisected at the point of contact.
The area of the triangle cut off from the asymptotes by any tangent is from (iii) equal to ^xy sin o) ; or, since
^xy' = a* + \^y and sin to — -^ — , .^ > ^^ 2x0.2, of the triangle
is equal to ah.
154. When the angle between the asymptotes of an hyperbola is a right angle it is called a rectangular hyper- bola.
The angle between the asymptotes is equal to 2 tan"^- ,
and therefore when the angle is a right angle we have \> — a. On this account the curve is sometimes called an equilateral hyperbola.
THE HYPERBOLA. 161
155. The asymptotes and any pair of conjugate diame- ters of an hyperbola form a harmonic pencil.
The tangent at the extremity of any diameter of an hyperbola is parallel to the conjugate diameter; also [Art. 153], the portion of the tangent intercepted by the asymptotes is bisected at the point of contact. Hence [Art. 55] the pencil formed by the asymptotes and a pair of conjugate diameters is harmonic.
156. We may, as in the case of the ellipse, express the co-ordinates of any point on the hyperbola in terms of a single parameter. We may put a; = a sec 6, and y = h tan 0, since for all values of 6, sec^6 — tan'^^ =1.
If PI^ be the ordinate of any point P on the curve, and JS^Q be the tangent from N to the auxiliary circle ; then (7iV"= a sec ACQ. Hence ACQ is the angle 0.
157. The equation of an ellipse or hyperbola referred to a vertex as origin is found by writing x — a for ^ in the equation referred to the centre as origin. The equation will therefore be
|
(w_ |
-af a' |
^y^- -Â¥ |
= 1, |
|
x' a:' |
2x^ a |
= 0 |
or -r,±^ = 0 l).
aba ^ ^
Now, if the distance from the vertex to the nearer focus
remain fixed {d suppose), and the eccentricity become
unity, the curve will become a parabola of latus rectum 4c?.
The equation of the parabola can be deduced from (i).
For, since a{l—e)=d, a must be infinite when e = l.
Also a{l-e'')=d{l + e) = 2d', therefore - = 2d. Hence, from (i)
or, since a is infinite,
a 2d
'if = ±Adx. s. c. s. 11
162 THE HYPERBOLA.
The parabola therefore is a limiting form of an ellipse or of an hyperbola, the latus rectum of which is finite, but the major and minor axes are infinite. The centre and the second focus are at infinity.
It is a very instructive exercise for the student to deduce the properties of a parabola from those of an ellipse or hyperbola.
158. Let the focus of a conic be on the directrix.
Take the focus as origin, and let the directrix be the axis of y; then the equation of the conic will be
or x^ (1 - e^) + 2/' = a
This equation represents two straight lines which are real if e be greater than unity, coincident if e be equal to unity, and imaginary if e be less than unity.
Hence we must not only consider as conies an ellipse, a parabola, and an hyperbola, but also two real or imaginary straight lines.
It should be noticed that the directrix of a circle is at an infinite distance ; also that the foci and directrices of two parallel straight lines are all at infinity.
Examples on Chapter VII.
1. A OB, COD are two straight lines which bisect one another at right angles; shew that the locus of a point which moves so that PA . PB = PC . PD is a rectangular hyperbola.
2. If a straight line cut an hyperbola in Q, Q' and its asymptotes in i?, R\ shew that the middle point of QQ' will be the middle point of RR\
3. A straight line has its extremities on two fixed straight lines and passes through a fixed point ; find the locus of the middle point of the line.
THE HYPERBOLA. 163
4. A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area ; find the locus of the middle point of the line.
5. OA, OB are fixed straight lines, F any point, and FM, FN the perpendiculars from P on OA, OB ; find the locus of F if the quadrilateral OMFN be of constant area.
6. The distance of any point from the centre of a rectan- gular hyperbola varies inversely as the perpendicular distance of its polar from the centre.
7. FN is the ordinate of a point F on an hyperbola, FG is the normal meeting the axis in G ; if NF be produced to meet the asymptote in Q^ prove that QG is at right angles to the asymptote.
8. If e, e be the eccentricities of an hyperbola and of the conjugate hyperbola, then will -g + -73 =-- 1.
Q Q
9. The two straight lines joining the points in which any two tangents to an hyperbola meet the asymptotes are parallel to the chord of contact of the tangents and are equidistant from it.
10. Prove that the part of the tangent at any point of an hyperbola intercepted between the point of contact and the transverse axis is a harmonic mean between the lengths of the perpendiculars drawn from the foci on the normal at the same point.
11. If through any point 0 a line OFQ be drawn parallel to an asymptote of an hyperbola cutting the curve in F and the polar of 0 in (^, shew that F is the middle point of OQ,
12. A parallelogram is constructed with its sides parallel to the asymptotes of an hyperbola, and one of its diagonals is a chord of the hyperbola; shew that the direction of the other will pass through the centre.
13. A^ A' are the vertices of a rectangular hyperbola, and F is any point on the curve; shew that the internal and external bisectors of the angle AFA' are parallel to the asymptotes.
11—2
164 THE HYPERBOLA.
14. A, A' are the extremities of a fixed diameter of a circle and F, F are the extremities of any chord perpendicular to this diameter ; shew that the locus of the point of intersec- tion of AF and A'F' is a rectangular hyperbola.
15. Shew that the co-ordinates of the point of intersection of two tangents to an hyperbola referred to its asymptotes as axes are harmonic means between the co-ordinates of the points of contact.
16. From any point of one hyperbola tangents are drawn to another which has the same asymptotes; shew that the chord of contact cuts off a constant area from the asymptotes.
17. The straight lines drawn from any point of an equi- lateral hyperbola to the extremities of any diameter are equally inclined to the asymptotes.
18. The locus of the middle points of normal chords of the rectangular hyperbola x^ — y^ = a' is (y^ - afy = ia^afy'.
1 9. Shew that the line x = 0 is an asymptote of the hyperbola 2xy + 3x' -f 4ic = 9.
What is the equation of the other asymptote 1
20. Find the asymptotes of ocy — 3x — 2y = 0. What is the equation of the conjugate hyperbola ?
21. Shew that in an hyperbola the ratio of the tangents of half the angles which the radii vectores from the foci to a point on the curve make with the axis, is constant.
22. A circle intersects an hyperbola in four points ; prove that the product of the distances of the four points of inter- section from one asymptote is equal to the product of their distances from the other.
23. Shew that if a rectangular hyperbola cut a circle in four points the centre of mean position of the four points is midway between the centres of the two curves.
24. If four points be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two, and if a, ft, y, 8 be the inclinations
THE HYPERBOLA. 165
to either asymptote of the straight lines joining these points respectively to the centre ; prove that tan a tan y8 tan y tan 8=1.
25. A series of chords of the hyperbola -g - Tg = ^ ^^®
Oj 0
tangents to the circle described on the straight line joining the foci of the hyperbola as diameter; shew that the locvis of
their poles with respect to the hyperbola is — 4 + ^ = -2 — j-^ .
26. If two straight lines pass through fixed points, and the bisector of the angle between them is always parallel to a fixed line, prove that the locus of the point of intersection of the lines is a rectangular hyperbola.
27. Shew that pairs of conjugate diameters of an hyperbola are cut in involution by any straight line.
28. The locus of the intersection of two equal circles, which are described on two sides AB, AG of a triangle as chords, is a rectangular hyperbola, whose centre is the middle point of BC, and which passes through A, B, C.
CHAPTER VIIL
PoLAK Equation of a Conic, the Focus being the
Pole.
159. To find the polar equation of a conic, the fociLS being the pole.
Let >S' be the focus and Z3f the directrix of the conic, and let the eccentricity be e.
Draw SZ perpendicular to the directrix, and let SZ be taken for initial line.
Let LSL' be the latus rectum, then e . SZ = SL = I suppose.
POLAR EQUATION OF A CONIC. 167
Let the co-ordinates of any point P on the curve be r, 0. Let PM, PN be perpendicular respectively to the directrix and to SZ.
Then we have
SP=:e.PM = e.NZ=^e.N8^e.SZ, or r = e . r cos (tt — ^) + / ;
^ V .• . - = 1 + e cos 6. r
If the axis of the conic make an angle a with the
initial line the equation of the curve will be
- = 1 + e cos (^ — a).
For in this case SP makes with SZ an angle B - a.
160. If r, 6 be the co-ordinates of any point on the directrix, then
rcosl9 = >SfZ=-; ' e
therefore the equation of the directrix is
- = e cos 6.
r
The equation of the directrix of - = 1 + e cos ^ — a is -= ecos (6 — a). ^
161. To shew that in any conic the semi-latus rectum is a harmonic mean between the segments of any focal chord.
If PSP' be the focal chord, and the vectorial angle of P be d, that of P' will be ^ -I- tt.
Hence, if 8P = r, and SP' = /, we have
- = 1 -I- e cos ^, and — = 1 + e cos (^ + tt) ; r r \ / >
.-. ^4 = 2. r r
TT 112
Hence - -f - = - .
r r L
168 POLAE EQUATION OF A CONIC.
162. To trace the conic - = 1 + ecos ^ from its equation. r
(1) Let e=l, then the curve is a parabola, and the equation becomes
- = 1 + cos e.
At^e point J., where the curve cuts the axis,
I
0=0 and r •
2'
As the angle 6 increases, (1 + cos 6) decreases, that is
- decreases, and therefore r increases : and r increases r
without limit until d = 7r, when r is infinite. As 6 in- creases beyond tt, 1 + cos 0 increases continuously, and therefore r decreases continuously until when 6 = 27r it
again becomes equal to ^ . The curve therefore is as in
the figure going to an infinite distance in the direction AS.
POLAR EQUATION OF A CONIC.
169
(2) Let e be less than unity, then the curve is an ellipse.
At the point A, ^ = 0, and r
l + e
As d increases cos^ decreases, and therefore - decreases,
r
that is r increases, until 6 = ir, when r = - — this value of r is positive.]
[Since e<l,
The curve therefore cuts the axis again at- some point
A' such that SA' = :. .
1 —e
As 9 passes from ir to 27r, cos 0 increases continuously from — 1 to 1 ; hence - increases continuously, and r de- creases continuously from to ^~— - .
Since, for any value of 6, cos 0 = cos (27r — 0), the curve is symmetrical about the axis.
Therefore when e is less than unity, the equation repre- sents a closed curve, symmetrical about the initial line.
(3) Let e be greater than unity, then the curve is an h3rperbola.
At the point A, 6 = 0 and r
1+e
170
POLAR EQUATION OF A CONIC.
As 6 increases cos 6 decreases, and therefore r increases until 1 i-e cos ^ = 0. For this value of 6, which we will call a (the angle ASK in the figure), the value of r will be infinitely great.
As 6 increases beyond the value a, (1 + e cos 6) becomes
I
negative, and when ^ = tt, r = —
e-1
SA' in the figure.
(1 +e cos 0) will remain negative until 6 is equal to (27r — a), the angle ASK' in the figure. When 6 is equal to (27r — a), r is again infinite. If 6 is somewhat less than this, r is very great and is negative, and if 0 is somewhat greater, r is very great and is positive. The values of r will remain positive while 6 changes from (27r — a) to 27r.
The curve is therefore described in the following order.
Fii-st the part ABC, then G'PA' and A'DE, and lastly E'QA,
The curve consists of two separate branches, and the radius vector is negative for the whole of the branch G'PA'DE.
If, as in the figure, a line SQP be drawn cutting the curve in the two points Q and P which are on different
POLAR EQUATION OF A CONIC. 171
branches, the two points Q and P must not be considered to have tlie same vectorial angle. The radius vector SP is negative, that is to say SF is drawn in the direction opposite to that which bounds its vectorial angle, the vectorial angle must therefore be ASp, p being on FS produced. So that, if the vectorial angle of Q be 6, that of P will be e-ir.
163. 2h find the polar equation of the straight line through two given points on a conic, and to find the equation of the tangent at any point.
Let the vectorial angles of the two points P, Q be (a — /3) and (a + ^) respectively.
Let the equation of the conic be
- = 1 + e cos 0 (i).
r
The straight line whose equation is
- = ^ cos ^ + P cos (^ — a) (ii),
will pass through any two points, since its equation con- tains the two independent constants A and B.
It will pass through the two points P, Q if r has the same values in (ii) as in (i) when ^ = a — y8, and when
This will be the case, if
1 + e cos (a — /3) = J. cos (a — /3) + P cos /S, and 1 +ecos(a +y8) = ^cos(a+yS) +PcosyS;
.•. A = e, and P cos /3 = 1. Substituting these values of A and B in (ii) we have the required equation of the chord, viz.
~ — e cos ^ + sec /3 cos {0 — a) (iii).
To find the equation of the tangent at the point whose vectorial angle is a, we must put /3 = 0 in (iii), and we obtain
- = ecos ^ + cos(^ — a) (iv).
172 POLAR EQUATION OF A CONIC.
Cor. If the equation of the conic be
- = 1 + e cos (^ — 7),
the chord joining the points (a — yS) and (a + ff) has for equation
- — e cos (^ — 7) + sec ^ cos (^ — a),
and the tangent at a has for equation
- = e cos (^ — 7) + cos (6 — a).
164. To find the equation of the polar of a point with respect to a conic.
Let the equation of the conic be
-= 1 + ecos^ (i),
r ^ [
and let the co-ordinates of the point be r^, d^.
Let a + yS be the vectorial angles of the points the tan- gents at which pass through {i\y 6^.
The equation of the line through these points will be
- = e cos ^ -I- sec /S cos (^ — a) (ii).
The equations of the tangents will be
- = e cos ^ + cos (^ — a + y8),
and - = eco^6 + cob{6 — a — P).
Since these pass through (r^ ^J, we have
- = e cos ^^ + cos (^j — a + /9) ;
and - = e cos ^j + cos {6^-a — fi) ;
r^
whence 6^ = a, and cos /3 = e cos 0^.
POLAR EQUATION OF A CONIC. 173
Substitute for a and ^ in (ii), and we have
ecos 6\ ( ecos^J — cos (6 — 6^,..(ui),
which is the required equation.
165. We will now solve some examples.
(1) The equation of the tangents at two points whose vectorial angles are a, /3 respectively are
- = e cos ^ + cos (^ - a), and - = 6Cos^ + cos (^-/3).
r
Where these meet
co8(^-a) = cos(^-/S);
Hence, if T be the point of intersection of the tangents at the two points P, Q of a conic, ST will bisect the angle PSQ. If however the conic be an hyperbola, and the points be on different branches of the curve, ST will bisect the exterior angle PSQ ; for, as we have seen, the vectorial angle of P (if P be on the further branch) is not the angle which SP makes with SZy but the angle PS produced makes with SZ.
(2) If the tangent at any point P of a conic meet the directrix in K, then the angle KSP is a right angle.
If the vectorial angle of P be a, the equation of the tangent at P will be
-=gcos^ + cos (^-a).
This will meet the directrix, whose equation is l = erGOBd, where cos(^-a) = 0.
Hence, at the point K, ^ - a = ± - .
Therefore the angle KSP is a right angle.
(3) If chords of a conic subtend a constant angle at a focus, the tangents at the ends of the chord icill meet on a fixed conic, and the chord will touch another fixed conic.
Let 2j3 be the angle the chord subtends at the focus. Let a-p and a + iS be the vectorial angles of the extremities of the chord.
174 POLAR EQUATION OF A CONIC.
The equation of the chord will be
;=6Cos ^ + secy3cos(^-a),
r
Zcos/3 r
:«cos/3.cos ^+cos {9 -a) (i).
But (i) is the equation of the tangent, at the point whose vectorial
angle is a, to the conic whose equation is
Zcos/3 ^ — ^ = l + ecos/3. cos 6 (u).
Hence the chord always touches a fixed conic, whose eccentricity is e cos /3, and semi-latus rectum I cos j3.
The equations of the tangents at the ends of the chord will be
I
- = ecos ^ + cos(^-a + /3),
and - = <?cos ^ + cos (^ — o-^).
Both these lines meet the conic
I
~ = e cos 6 + cos 8
r ^
in the same point, viz. where 6 = a and - = e cos a + cos /3.
Hence, the locus of the intersection of the tangents at the ends of the chord is the conic
Zsec/3 r
Both the conies (ii) and (iii) have the same focus and directrix as the given conic.
(4) To find the equation of the circle circumscribing the triangle formed by three tangents to a parabola.
Let the vectorial angles of the three points Ay B, G he a, p, y respectively.
Let the equation of the parabola be
-=l + cos^. r
The equations of the tangents Q.t A, B, C respectively will be
I
-=cos ^ + cos (d-a), r ^ '
I
- = C08e + C08(^-j3), I
- = cos ^ + cos(^-7).
= l + esec/3.eo8^ (iii).
POLAR EQUATION OF A CONIC. 175
The tangents at B and C meet where
^ = h(P + y)> ^^^ •*• -; = 2cos^cos|. The tangents at C and ^ meet where
d = i^[y + a), and- = 2cos^cos^. And the tangents at A and JS meet where
Q=z\{a. + S)^ and - = 2coSkC08§.
By substitution we see that the three points of intersection are on the circle whose equation is
2 ""° 2 """ 2
The circle always passes through the focus of the parabola.
(5) To find the polar equation of the normal at any point of a conic, the focus being the pole.
I f ^ 0. ^ y
2 cos - cos ^ cos ~r ^
|
Let the equation of the conic be |
|
|
- = l + ecos^. |
|
|
The equation of the tangent at any point a is |
|
|
- = ecos^ + cos(^-a). |
|
|
The equation of any line perpendicular to the tangent is ^=.COs(<»+|)+COB(9+|-a), |
|
|
- = -esin^-sin(0-a). |
or
This will be the required equation of the normal provided G is so
chosen that the point ( , ~ , a ) may be on the line. Hence we
^ \l + ecosa / "^
must have
_ 1 + e cos a
C =-e sm a,
- lesina ~l + e cos a *
Hence the equation of the normal is
Zesina 1 . „ . ,„
. -=«sm^ + sm {d-a).
1 + ecosa r
176 POLAR EQUATION OF A CONIC.
Examples on Chapter YIII.
1. The exterior angle between any two tangents to a parabola is equal to half the diflference of the vectorial angles of their points of contact.
2. The locus of the point of intersection of two tangents to a parabola which cut one another at a constant angle is a hyperbola having the same focus and directrix as the original parabola.
3. If PSP' and QSQ' be any two focal chords of a conic at right angles to one another, shew that ^^ — ^^^ + j^o — o7v is constant.
4. 11 A, £, C be any three points on a parabola, and the tangents at these points form a triangle A'B'C, shew that SA.SB.SG=SA'.SB'.SC', S being the focus of the pai-a- bola.
5. If a focal chord of an ellipse make an angle a with the axis, the angle between the tangents at its extremities is
_, 2e sin a
6. By means of the equation - = 1 + e cos ^, shew that the
ellipse might be generated by the motion of a point moving so that the sum of its distances from two fixed points is constant.
7. Find the locus of the pole of a chord which subtends a constant angle (2a) at a focus of a conic, distinguishing the cases for which cos a > = < e.
8. PQ is a chord of a conic which subtends a right angle at a focus. Shew that the locus of the pole of PQ and the locus enveloped by PQ are each conies whose latera recta are to that of the original conic as J2 : 1 and 1 : ^^2 respectively.
9. Given the focus and directrix of a conic, shew that the polar of a given point with respect to it passes through a fixed point.
POLAR EQUATION OF A CONIC. 177
10. If two conies have a common focus, shew that two of their common chords will pass through the point of intersection of their directrices.
11. Two conies have a common focus and any chord is drawn through the focus meeting the conies in P, P' and Q, Q' respectively. Shew that the tangents at P or P' meet those at Qy Q' in points lying on two straight lines through the inter- section of the directrices, these lines being at right angles if the conies have the same eccentricity.
12. Through the focus of a parabola any two chords LSL\ MSM' are drawn; the tangent at L meets those at M, M' in the points iV, N' and the tangent at L' meets them in K\ K, Shew that the lines KN^ K'N' are at right angles.
13. Two conies have a common focus about which one is turned; shew that two of their common chords will touch conies having the fixed focus for focus.
1 4. Shew that the equation of the locus of the point of intersection of two tangents to - = 1 + e cos 6, which are at right angles to one another, is i^ {f — \) — 2le r cos 6 + W= 0.
15. If PSQ, PHR be two chords of an ellipse through the
PS PH foci S, H, then will ^-pr + ^jtj be independent of the position
of P.
16. Two conies are described having the same focus, and the distance of this focus from the corresponding directrix of each is the same ; if the conies touch one another, prove that twice the sine of half the angle between the transverse axes is equal to the difference of the reciprocals of the eccentricities.
17. A circle of given, radius passing through the focus of a given conic intersects it in ^, By C, D; shew that
SA.SB.SC . SD
is constant.
s. c. s. 12
378 POLAR EQUATION OF A CONIC.
18. A circle passing througli the focus of a conic whose latua rectum is 21 meets the conic in four points whose distances
from the focus are r,, r , r.^, r ; prove that — H 1-— + — =^.
'\ '2 '3 'i ^
19. A given circle whose centre is on the axis of a parabola passes through the focus S, and is cut in four points A, B, C, D by any conic of given latus rectum having ^S* for focus and a tangent to the pai-abola for directrix ; shew that the sum of the distances SA^ SB, SC, SD is constant.
20. Two points P, Q are taken one on each of two conies, which have a common focus and their axes in the same direction, such that PS and QS are at right angles, S being the common focus. Shew that the tangents at P and Q meet on a conic the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original conies.
21. A series of conies are described with a common latus rectum; prove that the locus of points upon them, at which the perpendicular from the focus on the tangent is equal to the semi-latua rectum, is given by the equation ^ = - r cos 2^.
22. If POP' be a chord of a conic through a fixed point 0, then will tan \P'SO taxi ^P SO be constant, S being a focus of the conic.
23. Conies are described with equal latera recta and a common focus. Also the corresponding directrices en- velope a fixed confocal conic. Prove that these conies all touch two fixed conies, the reciprocals of whose latera recta are the sum and difference respectively of those of the variable conic and their fixed confocal and which have the same directrix as the fixed confocal.
;
CHAPTER IX.
GENERAL EQUATION OF THE SECOND DEGREE.
166. We have seen in the preceding Chapters that the equation of a conic is always of the second degree : we shall now prove that every equation of the second degree represents a conic, and shew how to determine from any such equation the nature and position of the conic which it represents.
167. To shew that evert/ curve whose equation is of the second degree is a conic.
We may suppose the axes of co-ordinates to be rect- angular; for if the equation be referred to oblique axes, and we change to rectangular axes, the degree of the equa- tion is not altered [Art. 53].
Let then the equation of the curve be
ax^ + 2hxy -H bf + 2ga; + 2ji/ + c = 0 (i).
As this is the most general form of the equation of the second degree it will include all possible cases.
We can get rid of the term containing oor/ by turning the axes through a certain angle.
For, to turn the axes through an angle ^ we have to substitute for x and y respectively xco^d — y sin 6, and xmiO + y cos 0 [Art. 50].
12—2
180 EVERY CURVE OF THE SECOND DEGREE IS A CONIC.
The equation (i) will become a (x cos ^ — y sin Q)^ + 2/^ {x cos ^ — 3/ sin 0) (x siiQ.d-{-y cos 6) +b(xsin0-}-ycos6f+2g(xcos6—ysmd)+2f{x8iD.d-hycos6) + c = 0 (ii).
The coefficient of xy in (ii) is
2 (6 - a) sin ^ cos ^ + 2/t (cos'^ - sin'^) ; and this will be zero, if
tan 2^ = —^ (iii).
Since an angle can be found whose tangent is equal to
any real quantity whatever, the angle 0 =^\ tan~^ 7^ is in
all cases real.
Equation (ii) may now be written
^ar' + %'4-2fe + 2i?V+ (7=0 (iv).
If neither A nor B be zero, we can write equation (iv) in the form
,( GV ^( FV GP" F' ^
— -J, — ^j ,
Aa^ + Bf = ^ + ^-C (v).
If the right side of (v) be zero, the equation will repre- sent two straight lines [Art. 35].
If however the right side of (v) be not zero, we have the equation
or
f
A\A^B-'^> b\a^-B-^)
which we know represents an ellipse if both denominators are positive, and an hyperbola if one denominator is posi- tive and the other negative.
If both denominators are negative, it is clear that no real values of x and of y will satisfy the equation. In this case the curve is an imaginary ellipse.
;
CENTRE OF A CONIC. 181
Next let A or B he zero, A suppose. [A and B cannot both be zero by Art. 53.] Equation (iv) can then be written
B(y + ^J = -^Gx-C + ~ (vi).
If (t = 0, this equation represents a pair of parallel straight lines.
If G be not zero, we may write the equation / Fv'_ 2Gf F^ G\ V'^BJ ~ B V~2BG'^2GJ' which represents a parabola, whose axis is parallel to the axis of X.
Hence in all cases the curve represented by the general equation of the second degree is a conic.
168. To find the co-ordinates of the centre of a conic.
We have seen [Art. 109] that when the origin of co- ordinates is the centre of a conic its equation does not contain any terms involving the first power of the variables. To find the centre of the conic, we must therefore change the origin to some point {x, y'), and choose x\ y\ so that the coefficients of x and y in the transformed equation may be zero.
Let the equation of the conic be
ax' 4- 2hxy + by^ + 2gx -\-2fy + c = 0.
The equation referred to parallel axes through the point {x, y) will be found by substituting x + x for x, and y -\- y for y, and will therefore be a{x + xy-\-2h{x + x')(^ + y') + b{y + yy + 2g{x+x)
or ax^ + 2hxy + by'' + 2x {ax + hy + g) + 2y {hx + by' + f) + ax"" + 'ihxy + by"" + igx' + 2fy + c = 0. The coefficients of x and y will both be zero in the above, if x and y' be so chosen that
ax+hy'-\-g = 0 (i),
and hx +by +/= 0 (ii).
The equation referred to (x', y) as origin will then be ax'' -h 2hxy -\-by^ + c ==0 (iii),
182 THE DISCRIMINANT.
where c' = aoi^'^ + llixy + hy"^ + Igx + Ify + c (iv).
Hence the co-ordinates of the centre of the conic are the values of a;' and y given by the equations (i) and (ii). The centre is therefore the point
~ah-h^' ah-K'J'
When ab — h^ = 0, the co-ordinates of the centre are infinite, and the curve is therefore a parabola [Art. 157]. If however hf— hg=0 and ah —h^ = 0 ; that is, if a_h_g
the equations (i) and (ii) represent the same straight line, and any point of that line is a centre. The locus in this case is a pair of parallel straight lines.
In the above investigation the axes may be either rectangular or oblique.
Subsequent investigations which hold good for oblique axes will be distinguished by the sign (©).
169. Multiply equations (i) and (ii) of the preceding Article by x' , y respectively, and subtract the sum from the right-hand member of (iv) ; then we have
_ hf— hg j,gh — o,f
ahc-^'lfgh-aP-hg^-ch^ , .
= ah^^W '• ("^)-
170. The expression aihc -f- ^fgli - af^ — hg^ — ch^ is usually denoted by the symbol A, and is called the discriminant of
ax"" + Vixy + hf -f- 2gx + 2/y + c.
A = 0 is the condition that the conic may be two straight lines.
For, if A is zero, c is zero ; and in that case equation (iii) Art. 168 will represent two straight lines.
This is the condition we found in Art 37. (&>).
AXES OF A CONIC. 183
171. To find the position and magnitvde of the axes of the conic whose equation is ao^ + ^hxy 4- hy^ = 1.
If a conic be cut by any concentric circle, the diameters through the points of intersection will be equally inclined to the axes of the conic, and will be coincident if the radius of the circle be equal to either of the semi-axes of the conic.
Now the lines through the origin and through the points of intersection of the conic and the circle whose equation is ^^ +3/^ = r^ are given by the equation
(a-i)^+2Aa;3,+ (6-J)/ = 0 (i).
These lines will be coincident, if
(»-?)('' -?)-'^'=" <"
and they will then coincide with one or other of the axes of the conic.
Hence the lengths of the semi-axes of the conic are the roots of the equation (ii), that is of the equation
^4-(« + &)^+a^-/^' = 0 .....(iii).
Multiply (i) by ia — ^2); then, if —^ is either of the roots of the equation (ii), we get
whence ia — 2) ^+ hy = 0 (iv).
Hence if we substitute in (iv) either root of the equation (iii) we get the equation of the corresponding axis.
In the above we have supposed the axes to be rect- angular. If however they are inclined at an angle co the investigation must be slightly modified, for the equation of the circle of radius r will be x^ -f- 2xi/ cos o) -I- ;/ = r"^.
184 AXIS OF A PARABOLA.
172. To find the axis and latus rectum of a parabola. If the equation
ax^ + '^.hxy + by" + 2^j; + 2/y + c = 0 represent a parabola, the terms of the second degree form a perfect square. [This follows from the fact that the equation of any parabola can be expressed in the form y"^ — 4>a'x — 0, and therefore with any axes the equation will be of the form
(Ix + my + n)^ — ^a {Vx + ni'y -hn) = 0.] Hence the equation is equivalent to
(ax + ^yy + 2gx + 2fy + c = 0 (i),
where a^ = a, and fi^ = b.
From (i) we see that the square of the perpendicular on the line ax + ffy = 0 varies as the perpendicular on the line 2gx + 2Jy + c = 0. These lines may not be at right angles, but we may write the equation (i) in the form
{ax + ^y + xy =:^2x (Xa- g) + 2y {\,8 -f) +\' - c, and the two straight lines, whose equations are ax + /3y + X=0, and 2x {Xa- g) + 2y (XjS -f) +X^- c = 0, will be at right angles to one another, if
a{Xa-g)+l3(Xfi-f)=0,
or if X = «4^/.
Now take ax-\-fiy+X=OsLnd 2(aX - g) x + 2 (^X-f) y -\-X^ - c = 0 for new axes of x and y respectively, and we get
y^ = 4!px, and this we know is the equation of a parabola referred to its axis and the tangent at the vertex.
To find the latus-rectum, we write the equation in the form
(ax + 0y + x\' {2{aX-ff)a:+2{fiK-f)y+\'-c] .
EXAMPLES OF CONICS. 185
hence ip^:^^-^^^^^-/!} .
Hence (i) is a parabola whose axis is the line ax + 0i/ + X = O, and whose latus-rectum is
2 V{(aX ~ gy + (/3\ -ff} _ 2 (af- ^g)
since \= ^,^^, .
173. We will now find the nature and position of the conies given by the following equations.
(1) 7a;2 - 11 xy + 6^/'' + 23a; - 2?/ - 20 = 0.
(2) a;^-5aj?/ + ?/2 + 8a;-20y + 15 = 0.
(3) 36a;2 + Uxy + 292/2 - 72a; + 126?/ + 81 = 0.
(4) (5a; - 12y)2 - 2a; - 29y - 1 = 0.
(1) The equations for finding the centre are [Art, 168, (i), (ii)]
14a;'- 17?/' + 23 = 01 - 17a;' + 12i/'- 2 = 01 These give re' = 2, 2/' = 3. Therefore centre is the point (2, 3).
The equation referred to parallel axes through the centre will be [Art. 169]
7x2 - 17.T?/ + 6i/2 + -^ . 2 - 1 . 3 - 20 = 0.
or 7a;2-l7a;?/ + 6i/2 = 0.
The equation therefore represents two straight lines which intersect in the point (2, 3). They cut the axis of x, where 7a;2 + 23a;- 20=0, that
is where a; = - 4, and where x — -.
(2) .'r2_5a;i^ + y2 + 8a;-202/ + 15 = 0. The equations for finding the centre are
2a;' - 52/' + 8 = 0, and - 5a;' + 2?/' - 20 = 0 ; .-. a;' = -4, 2/' = 0. The equation referred to parallel axes through the centre will be a;2 - 5a:2/ +?/2 + 4 ( - 4) + 15 = 0, or a;2 - ^xy + ?/2 = 1.
186
EXAMPLES OF CONICS.
The semi-axes of the conic are the roots of the equation 1
^-{a + b)\ + ab-h^ = 0 [Art. 171, (iii)];
1 2 , 25 ^
21r4 + 8r2-4 = 0;
2
2
.•.r^=7,or--. The curve is therefore an hyperbola whose real semi-axis is - ^/H, and whose imaginary semi-axis is - ^- 6.
\ I 1
The direction of the real axis is given [Art. 171, (iv)] by the equation
or x + t/=0.
(3) 36a:* + 24xy + 29y^ - 72a; + 126y + 81 = 0. The equations for finding the centre are
36a;' + 12/ - 36 = 0, and 12a/ + 29y' + 63 = 0 ;
.-. a;'=2, y' = -3.
The equation referred to parallel axes through the centre, will be
36a:2 -H 24a;y + 292/2 - 72 + 63 ( - 3) + 81 = 0,
x' 2 29 „ ,
or
EXAMPLES OF CONICS.
187
The semi- axes of the conic are the roots of the equation
And
, 65 13
ah-h^--
29
1^2..
900 225" 36' .-. 36 - 137-3 -i-r4 = 0. Hence the squares of the semi-axes are 9 and 4. Y
The equation of the major axis is [Art. 171, (iv)]
1
x + ~y = 0,
or 4x + 3y = 0.
(4) (5a;-12y)2-2a:-29i/-l=:0. The equation may be written
(5a; - 12y + X)2 = 2a; (1 + 5\) + 2/ (29 - 24X) + X2 + 1.
188 ASYMPTOTES.
The lines ox -12y + \=0
and 2(l + 5X)a;+(29-24X)y + X2 + l=:0
are at right angles, if
10 + 50\-348 + 288\ = 0; that is, if X = l.
The equation is therefore equivalent to
/5a;-12y+lY _ 1^ \ 13 / ~13
12a;+5y + 2
13 ^^''*
therefore 5a;- 122/ + 1=0 is the equation of the axis of the parabola, and 12a; + 5y + 2 = 0 is the equation of the tangent at the vertex.
Every point on the curve must clearly be on the positive side of the line 12x-\-5y + 2 = 0, since the left side of equation (i) is always positive.
174. To find the equation of the asymptotes of a conic.
We have seen [Art. 146] that the equations of a conic and of the asymptotes only differ by a constant.
Let the equation of a conic be
aa;'+ 2hxy+bf + 2ga) + 2fy + c=0 (i).
Then the equations of the asymptotes will be
aa^+2hxy + hf + 2gx + 2fi/+c + X=^0 (ii),
provided we give to \ that value which will make (ii) represent a pair of straight lines.
The condition that (ii) may represent a pair of straight lines is [Art. 170]
ab{c-\-X) + 2fgh-aP - Ig^ - (c + X) /i= = 0 ; ... x(a6-/^') + A = 0. Hence the equation of the asymptotes of (i) is
ax^ + 2hxy + hy" + 2gx + 2/y + c - . _^2= 0.
The equations of two conjugate hyperbolas differ from the equation of their asymptotes by constants which are equal and opposite to one another [Art. 152] ; therefore the equation of the hyperbola conjugate to (i) is
2A
ax" â– ^2hxy â– \-hy'' â– \-2gx â– \-2fy -^ c 7-3^72 = 0.
RECTANGULAR HYPERBOLA. 189
Cor. The lines represented by tlie equation ax' + 2hxy + 63/^ = 0 are parallel to the asymptotes of the conic, (&>).
Ex. Find the asymptotes of the conic
The asymptotes will he x^-xy-'iy^-^iy-2+\ = Q, if this equation represents straight lines. Solving as a quadratic in x, we have
Hence [Art. 37], the condition for straight lines is 9 (2 - X) = 9, or X = 1. The asymptotes are therefore x^-xy- 2y^ + Sy — 1=0.
175. To find the condition that the conic represented hy the general equation of the second degree may he a rect- angular hyperbola.
If the equation of the conic be
ax' + '^hxy + hy^ + ^x + 2/?/ + c = 0, the equation
ax'' + 2hxy + hy'' = 0 (i)
represents straight lines parallel to the asymptotes.
Hence, if the conic is a rectangular hyperbola, the lines given by (i) must be at right angles.
The required condition is therefore [Art. 44]
a + 6 — 2^ cos ft) = 0 (ii).
If the axes of co-ordinates be at right angles to one another the condition is
a + 6 = 0 (iii).
The required condition may also be found as follows. If the axes of co-ordinates be changed in any manner whatever, we have
a + 6 — 2^cosft> a' + 6' — 2/i.' cos ft)' . , ^^-,
T-^ = r-^— 7 [Art. 52].
sm ft) sm ft) '- -^
But, if the conic be a rectangular hyperbola and
190 EXAMPLES.
the asymptotes be taken for axes, the equation will be xy + constant = 0 ;
.*. a' = 6' = cos a>' = 0.
Hence a + 6 — 2/t cos &> = 0. («).
»
Examples on Chapter IX.
1. Find the centres of the following curves: (i) Zx^ - ^xij + 6/ + llaj - 17y + 13 = 0.
(ii) xy + 3aic - Say = 0. (iii) 3x^-7a?/-62/' + 3a;-92/ + 5 = 0. Find also the equations of the curves referred to parallel axes through their centres.
2. What do the following equations represent ?
(i) xy-2x + y-1 = 0. (ii) y^ - 2ay + 4:ax - 0. (iii) y^ + ax + ay + a^ = 0. (iv) (x + yY ==a{x-y). (v) 4:(x+2yy+{y-2xy^5a\ (vi) y'-x^-2ax = 0.
3. Draw the following curves:
(1) xy + ax-2ay = 0. (2) of + 2xy+y' - 2x-l =^0.
(3) 2x' + 5xy-^-2y' + 3y-2:^0.
(4) x' + 4:xy + y'-U = 0.
(5) (2x + 3yy + 2x+2y+2==0.
(6) x^-4:xy-2y^-i-10x + 4:y = 0.
(7) Ux'+ 2ixy + V - 130aa; - eOay + 116a" = 0.
4. Shew that if two chords of a conic bisect each other, their point of intersection must be the centre of the curve.
5. Shew that the product of the semi-axes of the conic whose equation is
(a; - 22/ + 1)'+ (4a; +22/- 3)" -10 = 0, is 1.
6. Shew that the product of the semi-axes of the ellipse whose equation is
x' -xy + 2y' -2x- 6y + 7 ^ 0 is -^ ;
and that the equation of its axes is
x^ - y^ - 2xy + 8y - 8 =^0.
EXAMPLES. 191
7. Find for what value of A. the equation
2a;' + Xa;?/-2/' - 3£c + 6y - 9 = 0 will represent a paii' of straight lines.
8. Find the equation of the conic whose asymptotes are the lines 2a; + 3y — 5 = 0 and 5x + Sy - 8 = 0, and which passes through the point (1, — 1).
9. Find the equation of the asymptotes of the conic
3a;" - 2xy-5y' + 7x-9y = 0; and find the equation of the conic which has the same asymp- totes and which passes through the point (2, 2).
10. Find the asymptotes of the hyperbola
6x' -7xy-3y'-2x-Sy-6 = 0; find also the equation of the conjugate hyperbola.
11. Shew that, if
aaf + 2hxy + by^ = 1, and aV + 2hfxy + b'y^ = 1 represent the same conic, and the axes are rectangular, then
(a-by + ih' = {a'-hy + 4.h".
12. Shew that for all positions of the axes so long as they remain rectangular, and the origin is unchanged, the value of g^ + /^ in the equation ax^ + 2hxy + by^ + 2gx + 2fy + c = 0 is constant.
13. From any point on a given straight line tangents are drawn to each of two circles: shew that the locus of the point of intersection of the chords of contact is a hyperbola whose asymptotes are perpendicular to the given line and to the line joining the centres of the two circles.
14. A variable circle always passes through a fixed point 0 and cuts a conic in the points P, Q, R, S; shew that
OP. OQ. OB. OS (radius of circle)* is constant.
15. If ao(f + 2Iixy + by^=l, and Aaf +2Hxy + Bf = 1 be the equations of two conies, then will aA +bjB + 2hU be un- altered by any change of rectangular axes.
CHAPTER X.
MISCELLANEOUS PROPOSITIONS.
176. We have proved [Art. 167] that the curve represented by an equation of the second degree is always a conic.
We shall throughout the present chapter assume that the equation of the conic is
aa? + 2hxy + hif + 2gx + 2/^/ + c = 0, unless it is otherwise expressed.
The left-hand side of this equation will be sometimes denoted by <f> {x, y),
177. To find the equation of the straight line passing through two points on a conic, and to find the equation of the tangent at any point.
Let {x, y') and (x', y") be two points on the conic.
The equation a(x-x'){x-x") + h[{x-x')(:y-y") + {x-x"){y-y')] + hiy-y'){y-y")=ax' + 2hxy + h/-{-2gx-h2fy + c..,(i)
when simplified is of the first degree, and therefore represents so7ne straight line.
EQUATION OF A TANGENT. 193
If we put x = x and y — y in (i) the left side vanishes identically, and the right side vanishes since {x , y) is on the conic. Hence the point {x , y') is on the line (i). So also the point {x'\ y") is on the line (i).
Hence the equation of the straight line through the two points {x\ y) and {x", y") is (i), and this reduces to ax {x + x") + ky {x + x) + hx {y + y") + hy (y + y") + 2gx -f Ify + c = axx" 4- h {xy" + y'oc'^ + hy y" ... (ii).
To obtain the tangent at {x\ y) we put x" — x\ and y" = y in (ii), and we get 2axx + 2/t [xy + xy) + 26y?/' + ^x ^'^fy + c = ax""
+ 2hxy'-\-hy\
Add 2gx' + 2/y + c to both sides : then, since {x\ y') is on the conic, the right side will vanish ; and we get for the equation of the tangent axx + h (y'x + xy) + hyy +g{x + x) +f{y -{â– y')+c = 0.
It should be noticed that the equation of the tangent at {x, y) is obtained from the equation of the curve by writing xx for o^^ yx + x'y for 2xy, yy for y"^, x-\-x for 2^, and y -{-y for 2y. (<w).
178. To find the condition that a given straight line may he a tangent to a conic.
Let the equation of the straight line be
Ix + my + ?i = 0 (i).
The equation of the straight lines joining the origin to the points where the line (i) cuts the conic </> {x, y) = 0, are given [Art. 38] by the equation
aa? + tU^y + hf-2 (gw +fy) ^^±^
+ c('^y = 0 (ii).
If the line (i) be a tangent it will cut the conic in coincident points, and therefore the lines (ii) must be coincident. The condition for this is
s. C. s. 13
194 THE POLAR.
{an^ - 2gln + cl"^) (bn^ - 2f7nn + cm^)
= {hn^ ~"/^'^ —gmn H- clnif, or Z' {he -f) + 7?t' (m - /) + n' {ah - Jv") + 2m;i {gh -fa)
+ 271^ (/i/- ^6) + 2lm {fg-hc) = 0 (iii).
The equation (iii) may be written in the form AP + B'nv' + Gn^ + 2Fmn + 2 Gnl + 2Hlm = 0. . .(iv), where the coefficients A, B, (7, &c. are the minors of a, h, c, &c. in the determinant
a, h, g \ h.h,f '
Oy f> 0 1 (ft)).
179. I'o find the equation of the polar of any point with respect to a conic.
It may be shewn, exactly as in Article 76, 100, or 118, that the equation of the polar is of the same form as the equation of the tangent.
The equation of the polar of {x , y) is therefore axx + h {y'x + xy) + hy'y -\- g {x-V x) -^f(y +y') + c=0, or X (ax + hy' + g) -\- y (hx + by +/) + gx -\-fy' + c = 0.
The equation of the polar of the origin is found by putting ^' = y' = 0 in the above ; the result is gx-^fy+c = 0.
180. Jf two points P, Q he such that Q is on the polar of P with respect to a conic, then will P he on the polar of Q with respect to that conic.
Let the co-ordinates of P be x, y\ and those of Q x, y".
The equation of the polar of P is axx -f h {^x + xy) + hy'y -V g {x \ x) •\-f{y + y') + c = 0.
Since {x\ y") is on the polar of P, we have axx" + h {yx"-\-xy") + hyy" + g {x+x')+f(y'+ y") + c=0.
The symmetry of this result shews that it is also the condition that the polar of Q should pass through P.
THE POLAR. 195
If the polars of two points P, Q meet in R, then R is the pole of the line PQ.
For, since B is on the polar of P, the polar of R will go through P; similarly the polar of R will go through Q ; and therefore it must be the line PQ.
If any chord of a conic he drawn through a fixed point Q, and P be the pole of the chord ; then, since Q is on the polar of P, the point P will always lie on a fixed straight line, namely on the polar of Q.
Def. Two points are said to be conjugate with respect to a conic when each lies on the polar of the other.
Def. Two straight lines are said to be conjugate with respect to a conic when each passes through the pole of the other. Conjugate diameters, as defined in Art. 127, are conjugate lines through the centre.
181. If any chord of a conic be drawn through a point 0 it will he cut harmonically by the curve and the polar of 0.
Let OPQR be any chord which cuts a conic in P, R and the polar of 0 with respect to the conic in Q.
Take 0 for origin, and the line OPQR for axis of x ; and let the equation of the conic be
ax''+2hxy-\-by''^2gx + ^fy-{-c= 0.
Where y = 0 cuts the conic we have
ax^ + ^gx + c = 0 ;
1 1 __%
•'• OP^ OR c ^'^•
The equation of the polar of (9 is gx-\-fy + c = 0\
1 a ....
■-■Wq—Z (">•
From (i) and (ii) we see that
112
OP "^ OR OQ
13—2
196 CONJUGATE DIAMETERS.
182. To find the locus of the middle j^oints of a system of 'parallel chords of a conic.
Let {x\ y) and (x" , y") be two points on the conic. The equation
a{x- x) {x - x") + h{{x-x){y- y") â– \-{x- x") {y - y')} + h{y -y') {y-y") = aT^ + 2hxy+hf -V^gx + ^fy -\.c
..•(i). is the equation of the straight line joining the two points.
In (i) the coefficient of a; is a (x-\- x") + h(^y' + y') + ^y,
and the coefficient of y is h {x + x") + b (y -{- y") 4- 2/*;
hence if the Hne is parallel to the line y = mx, we have
a(a:'+x") + h(y' + f)+2g
- h (X + :r") + 6 (2,' + y") + 2/ ^"''-
Now, if {x, y) be the middle point of the chord joining
{x'y y') and {x" , y"), then ^x — x'-'r x\ and 2y = y' -\- y" ;
therefore, from (ii), we have
ax + hy -\- a
kx-hby+f
or X (a + mh) -h y {h + mh) + g -\- inf= 0. . .(iii ),
which is the required equation.
If the line (iii) be written in the form y = mx + k, then
we have
, _ a + mh
'^'^~h + mb'
or a-^h (m + m) 4- hmm^ = 0 (iv).
This is the condition that the lines y = m^ and y = mx
may be parallel to conjugate diameters of the conic given
by the general equation of the second degree. (to).
183. To find the condition that the two lines given by the equation Ax^ + 2Hxy + By"^ = 0 may be conjugate dia- Tneters of the conic ax^ + 2hxy + by^ — 1.
If the lines given by the equation Ax^-\-2Hxy-\-By'^—0 be the same as y — mx = 0, and y — mx = 0 ; then
on + m = — 2 ^ , and mm = ^ .
CONJUGATE Diameters. 197
But y — vix= 0 and y—mx=0 are conjugate diameters if a + h (m + m) + hmm = 0.
Therefore the required condition is
or aB + bA = 2hH. (a>).
[The above result follows at once from Articles 155 and 58.]
Ex. 1. To find the equation of the equi-conjugate diameters of tlie conic
ax^ + 2hxi/ + by^ = l. The straight lines through the centre of a conic and any concentric circle give equal diameters. Through the intersections of the conic and the circle whose equation is \ {x^ + y^ + 2xi/ cos w) = 1, the Hnes {a - X) x- + 2{h-\ cos w) xi/ + {b-\)y^ = 0 pass. These are conjugate if
h{a-\) + a [h - X) = 2/i {h - X cos w). Substituting the value of X so found, we have the required equation
ax^ + 2hxy + hy^ - , --^~ '' ' (x^ + y^ + 2xn cos w) = 0.
Ex. 2. To shew that any two concentric conies have in general one and only one pair of common conjugate diameters. Let the equations of the two conies be
ax^ + 2 hxy + hy^=l, and a'x^ + 2 li'xy + h'y^ = 1 . The diameters Ax^ + 2Hxy + By'^ = 0 are conjugate with respect to both conies if
Ab-2Hh + Ba = 0,
and Ab'- 2Hh' + 7?a' = 0 ;
A _ -2H _ B ha' - ah' ~~ ah' - a'b hh' - b'h ' The equation of the common conjugate diameters is therefore
{ha' - ah') x^ - {ab' - a'b) xy + {hh' - b'h) y^ = 0.
Since any two concentric conies have one pair of conjugate diameters in common, it follows that the equations of any two concentric conies can be reduced to the forms
ax'^ + by^ = lf a'x^ + b'y'-=r.l.
198 SEGMENTS OF A CHORD.
184. To find the length of a straight line drawn froTii a given point in a given direction to meet a conic.
Let {x, y') be the given point, and let a line be drawn through it making an angle 6 with the axis of x. The point which is at a distance r along the line from {x, y) is {x + r cos 6, y' + r sin 0\ the axes being supposed to be rectangular ; and, if this point be on the conic given by the general equation, we have a (.^•'+ r cos 6y-\-^h {x + r cos 6) {y' -\- r sin 6) + h (y'+rsin^)*
4- 2^ {x 4- r cos d) + 2/(2/' + r sin ^) + c = 0, or r' (a cos^ O^-^h sin 6cos6 + h sin' 6)
+ 2r cos 6 (ax -\- hy +^) + 2r sin 6 {hx -Vhy â– >rf)-^(^ {x , y) = 0.
The roots of this quadratic equation are the two values of r required.
185. If the point (x\ y') be the middle point of the chord intercepted by the conic on the line, the two values of r, given by the quadratic equation in the preceding Article, will be equal in magnitude and opposite in sign ; hence the coefficient of r must vanish ; thus
{ax + hy' + g) cos 6 + Qix + hy +/) sin ^ = 0.
If the chords are always drawn in a fixed direction, so that '6 is constant, the above equation gives us the relation satisfied by the co-ordinates x\ y of the middle point of any chord.
The locus of the middle points of chords of the conic which make an angle 0 with the axis of x is therefore a straight line. [See Art. 182.]
186. The rectangle of the segments of the chord which passes through the point {x\ y') and makes an angle 0 with the axis of x^ is the product of the two values of r given by the quadratic equation in Art. 184; and is equal to
a cos^ 6 + 2h sin 6 cos 6 + b sin^O '
Cor. 1. If through the same point {x, y') another chord be drawn making an angle ff with the axis of x, the
SEGMENTS OF A CHORD. 199
rectangle of the segments of this chord will be
</> {x', y)
a cos^ & + 2/fc sin & cos & ^-h sin^ & '
Hence we see that the ratio of the rectangles of the seg- ments of two chords of a conic drawn in given directions through the same point is constant for all points, including the centre of the conic, so that the ratio is equal to the ratio of the squares of the parallel diameters of the conic.
Gor. 2. The ratio of the two tangents drawn to a conic from any point is equal to the ratio of the parallel diame- ters of the conic.
Gor. 3. If through the point {x'\ y") a chord be drawn also making an angle 6 with the axis of x, the rectangle of the segments of this chord will be
</> [^"> y")
a cos^ 6 + 2h sin 6 cos 0 +h sin^ 0 '
Hence the ratio of the rectangles of the segments of any two parallel chords drawn through two fixed points (x, y) and {x\ y") is constant and equal to the ratio of ^ {x, y) to </) {x'\ y").
Got, 4. If a circle cut a conic in four points P, Q, ii, ^, the line PQ joining any two of the points and the line R^ joining the other two make equal angles with an axis of the conic.
For, if PQ and R^ meet in T, the rectangles TF .TQ and TK . T8 are equal since the four points are on a circle. Therefore by Cor. 1, the parallel diameters of the conic are equal ; and hence they must be equally inclined to an axis of the conic.
Ex. 1. If o, /3, 7, S he the eccentric angles of the four points of inter- section of a circle and an ellipse^ then will a + ^ + 7 + 5 = 27i7r. The equations of the lines joining a, /3 and 7, 5 are
^ cos |(a + /S) + 1 sin |(a + /3) = cos \{a-^),
200 CONICS HAVING FOUR COMMON POINTS.
and - cos ^ (7 + 5) +^ sin ^ (7+ 5) = cos ^ (7 - 5).
These two chords are equally inclined to the axis by Cor. 4 : therefore tan| (a + /3) = -tan^ (7 + 5), or -^ (a +^)^7/7r-^ (7 + 5); therefore a+p + y + 5 = 2nir.
Ex. 2. A focal chord of a conic varies as the square of the parallel diameter. [See Art. 161.]
Ex. S. If a triangle circumscribe a conic the three lines from the angular points of the triangle to tJie points of contact of the opposite sides will meet in a point.
Let the angular points be ^, J5, C and the points of contact of the opposite sides of the triangle be A', B\ C; also let r^, ro, r.^ be the semi- diameters of the conic parallel to the sides of the triangle. Then
BA' : BG'=r^ : r^; CB' : CA' = r^ : r^; and AC : AB' = r^ : r^. Hence BA' . CB' . AC'=BC. AB' . CA',
which shews that the three lines meet in a point.
Ex. 4. If a conic cut the three sides of a triangle ABC in the points A' and A'\ B' and B", C and G" respectively, tlien will
BA' . BA" . CB' . CB" . AC . AC" = BC . BC" . CA' . CA" . AB' . AB".
{CarnoVs Theorem.)
{BA' . BA" : BC . BC" = r{^ : r,,-, and so for the others ; r^, ?•.,, r.^ being the semi-diameters of the conic parallel to the sides of the triangle,]
Ex. 5. If a conic touch all the sides of a polygon ABCD the
jjoints of contact of the sides AB, BC being P, Q, R, S ; then will
AP.BQ.CR.DS be equal to PB .QC .RD
187. If >S^ be written for shortness instead of the left- hand side of the equation
aa^ + 2hxj/ + bf + 2ga; + 2fy + c = 0,
and S' be written instead of the left-hand side of the equation
aV-f 2h'a;i/ + by + 2(]'x + 2fy + c' = 0,
then S — \S' = 0 is the equation of a conic which passes through the points common to the two conies
For, the equation S — \8' = 0 is of the second degree, and therefore represents some conic. Also if any point be
CONICS HAVING DOUBLE CONTACT. 201
on both the given conies, its co-ordinates will satisfy both the equations >Sf = 0 and S' = 0, and therefore also the equation S—\S' = 0.
By giving a suitable value to \, the conic S — \S'=0 can be made to satisfy any one other condition.
If the conic S' = 0 really be two straight lines whose equations are Ix + my + ?i = 0 and I'co + m'y + n' = 0, which for shortness we will call u = 0, and v = 0, then S — \uv = 0 will, for all values of X, be the equation of a conic passing through the points where >S» = 0 is cut by the lines io = 0 and v = 0.
If now the line v = 0 be supposed to move up to and ultimately coincide with the line w = 0, the equation S — \u^ = 0 will, for all values of X, represent a conic which cuts the conic >Sf = 0 in two pairs of coincident points, where >Sf = 0 is met by the line u = 0. That is to say S—Xu^ = 0 is a conic touching /S'=0 at the two points where >S'= 0 is cut by u = 0. {co).
Ex. 1. All conies through the points of intersection of two rectangular hyperbolas are rectangular hyperbolas.
It S = 0, S' = 0 he the equations of two rectangular hyperbolas, all conies through their points of intersection are included in the equation S-\S' = 0. Now the sum of the coefficients of x^ and y^ in S-\S' = 0 will be zero, since that sum is zero in 8 and also in S\ the axes being at right angles. This proves the proposition. [Art. 175.]
The following are particular cases of the above.
(i) If two rectangular hyperbolas intersect in four points, the line joining any two of the points is perpendicular to the line joining the other two. (For the pair of lines is a conic through the points of intersection.) (ii) If a rectangular hyperbola pass through the angular points of a triangle it will also pass through the orthocentre. (For, if ^, £, Che the angular points, and the perpendicular from A on BC cut the conic in D ; then the pair of lines AD, J5C is a rectangular hy- perbola, since these lines are at right angles ; therefore the pair BD, AC is also a rectangular hyperbola, that is to say the lines are at right angles. )
E-x. 2. If tico conies Imve their axes parallel a circle will pass through their jioints of intersection.
202 EQUATION OF A PAIR OF TANGENTS.
Take axes parallel to the axes of the conies, their equations will then be ax- + hy"^ + 2gx + 2/^/ + c = 0,
and a'x^ + 6^ + 'ig'x + 2/y + c' = 0.
The conic ax"^ + h]f + ^gx + 2/y + c + X {a'x^ + h'lf + 2g'x + 2fy + c') = 0 will go through their intersections. But this will be a circle, if we choose X so that a + Xa' = & + X6', and this is clearly always possible.
Ex. 3. If TP, TQ and T'P', T'Q' he tangents to an ellipse, a conic tvillpass through the six points T, P, Q, T', P', Q'.
Let the conic be ax^ + by^ = l, and let T be {x\ y') and T' be (ac", y"). The equations of PQ and P'Q' will be axx' + hyy' -1 = 0 and ax3d' + hyy" -1 = 0. The conic
X [ax^ + &y2 _ 1) _ f^axx' + hyy' - 1) [axx" + hyy" - 1) = 0 will always pass through the four points P, Q, P', Q'. It will also pass through T if X be such that
X [ax"' + &y'2 - 1) - (ax 2 + 6i/'2 1) (oa/a;" + fty'y" - 1) = 0, or if X = flxV + ftyV - 1.
The symmetry of this result shews that the conic will likewise pass through T'.
Ex. 4. If two chords of a conic he drawn through two points on a diameter equidistant from the centre, any conic through the extremities of those chords will he cut hy that diameter in points equidistant from the centre.
Take the diameter and its conjugate for axes, then the equation of the conic will be ax'^ + hy" = l. Let the equations of the chords be y -m{x-c)=0 and y - m' {x + c) = 0. Then the equation of any conic through their extremities is given by
flx2 + &?/2_i_x {y-m{x-c)] {y~m'{x + c)}=0.
The axis of x cuts this in points given by aa;^- 1-Xmw' (a;2-c-) = 0, and these two values of x are clearly equal and opposite whatever X, m and m' may be.
As a particular ease, if PSQ and P'S'Q' be two focal chords of a conic, the lines PP' and QQ' cut the axis in points equidistant from the centre.
188. To find the equation of the pair of tangents drawn from any "point to a conic.
Let the equation of the conic be
ax^ + 2hxy + 6/ + %jx + 2fy + c = 0 (i).
If {oc\ y) he the point from which the tangents are drawn, the equation of the chord of contact will be
THE DIRECTOR CIRCLE. 203
axx' 4- h {xy -f yx'^ + hyy + ^ (ic -f x) â– \-f{y + y')-\-c = 0. The equation
ax" + ^hxy + hy'^ + 2gx + 2fy-\- c = X {a*"X'' + h [xy' + y^') + hyy' + g (x + x) +/ (?/ + y) + c}"
(ii)
represents a conic touching the original conic at the two points where it is met by the chord of contact. The two tangents are a conic which touches at these two points and which also passes through the point {x, y') itself. The equation (ii) will therefore be the equation required if X be so chosen that {x , y) is on (ii) ; that is, if ax"" + "Lhxy + hy^ + "Igx + 2// + c = X [ax"- + 2A^y + hy'^ + "Lgx + 2/y' + c\\ Therefore 1 = X [ax"" + "Lhxy' + hy'^ + "Igx + 2// + c} = X<^ (.^', y\ Substituting this value of X in (ii) we have {ax"- + "Ihxy + 6?/' + "Lgx + 2/3/ H- c) (jb (a?', y') = [axx + A (a??/' + yx') + ^y + ^ (^ + ^0 +/(y + 2/') + c}', which is the required equation. (w).
The above equation may be found in the following manner.
Let TQ, TQ' be the two tangents from (x\ if), let P {x, y) be any point on TQ, and let TN, PM be the perpendiculars from T and P on the chord of contact QQ'.
PQ^_PM^
But [Art. 186,00.3] ^ = ^p^%
and [Art. 31]
PM^ _ {axx' + h{xy' + y3r} + byy' + g{x + ccr)+f{y + y') + c}^ . TN^ ~ {ax'^ + 2hx'y' + 6y'2 + 2gx' + 2fy' + c } 2
therefore from (i) we have
0 (x, 2/) 0 (x', 2/') = {axx' + h {xy' + yx') + byy' + g{x + x') +f (y + y') + c\K
189. To find the equation of the director-circle of a conic.
The equation of the tangents drawn from (x\ y') to the conic given by the general equation is
Then i^, = ^ (i).
204 THE FOCI.
(ax^ + 2hxy + %^ + 2^^ + 2/1/ + c)(j> {x, y) = {axx' + A (^773/' + yx) '+ hyy +g{x-\- x) +f{y + /) + c}\
The two tangents will be at right angles to one another if the sum of the coefficients of x^ and y^ in the above equation is zero. This requires that
(a + b) («*•'' + 2hxy' + hy"" + Igx + Ify + c) - {ax + hy + ^)'^ - (^x + %' +/)' = 0.
The point {x , y') is therefore on the circle whose equation is
{ah - h') (x"" + f) + 2x {gh -fh) â– \-2y{fa- hg) ^c{a-\-h)
or Cx' + Gf-^Ox-2Fy+A+B=0 (i),
where A, B, G, F, G, H mean the same as in Art. 178,
If h^ — ab = 0, the equation reduces to
2x (hg -fh) + 2y (fa - hg) + c (a + h)-f -/ = 0,
or 2Gx-\-2Fy-A--B=0 (ii).
The conic in this case is a parabola, and (ii) is the equation of its directrix.
Ex. 1. Trace the curve llx^ + 24a;y f 4i/ -2x+l&y + ll=0, and shew that the equation of the director-circle is x^ + y^ + 2x ~ 2y = 1.
Ex. 2. Shew that the equation of the directrix of the parabola x^ + 2xi/ + 1/ - Ax + 8y- 6 = 0 is 3x-3y + S = 0.
190. To shew that a central conic has four and only four foci, tiuo of which are real and two imaginary.
Let the equation of the conic be
aa;' + 6/-l = 0 (i).
Let (x, y) be a focus, and let x cos a + y sin a—p = 0 be the equation of the corresponding directrix ; then if e be the eccentricity of the conic, the equation will be
(x — xy-\- {^ — yf — e*(^cos a +ysin a — p)* = 0...(ii).
THE FOCI. '205
Since (i) and (ii) represent the same curve, and the coefficient of xy is zero in (i), the coefficient of acy must be
zero in (ii) ; hence a is 0 or ^ .
Hence a directrix is parallel to one or other of the axes.
Let a = 0, then since the coefficients of x and y are zero in (i), we have y =0 and x — i'p.
Also, by comparing the other coefficients in(i) and (ii), we have
a^ _h^ -1
■•■-y(^-i) ("^)'
apx =1 (iv),
and x^ = -,- (v).
a b
From (v) we see that there are two foci on the axis of
X whose distances from the centre are ±a/( t) •
From (iv) we see that a directrix is the polar of the corresponding focus.
IT
If a = - , we can shew in a similar manner that there
are two foci on the axis of y whose distances from the
centre are ± a /(r ) • Of the two pairs of foci one is
clearly real and the other imaginary, whatever the values of a and h (supposed real) may be.
The eccentricity of a conic referred to a focus on the
axis of X is from (iii) equal to . /( 1 — ^ j; the eccentricity
referred to a focus on the axis of y will similarly be
1 — j . If the curve be an ellipse a and h have the
^/(
206 THE ECCENTRICITY.
same sign, and one of these eccentricities is real and the other imaginary. If however the curve be an hyperbola, a and h have different signs and both eccentricities are real.
In any conic, if e^ and e^ be the two eccentricities, we have
11a h ,
-2+ -Â¥ = r+7 =1-
e^ e^ a— 0 b — a
191. To find the eccentricity of a conic given by the genei^al equation of the second degree.
By changing the axes we can reduce the conic to the form
aaj' + yS/ + 7 = 0 (i).
If e be one of the eccentricities of the conic,
a = /S(l-e^)., (ii).
But [Art. 52], we know that
a + P = a + b (iii),
and afi — ab — h^ (iv).
Eliminating a and yS from the equations (ii), (iii) and (iv), we have
{2-ey ^{a + by l-e' ab-h''
^+ ab-k' (.'-1)=0 (V).
If the curve is an ellipse, ab — h^ is positive, and one value of e^ is positive and the other negative. The real value of e is the eccentricity of the ellipse with reference to one of the real foci, and the imaginary value is the eccentricity with reference to one of the imaginary foci.
If the curve is an hyperbola both values of e^ are positive, and therefore both eccentricities are real, as we found in Art. 190; we must therefore distinguish between the two eccentricities.
The signs of a and fi in (i) are different when the curve is an hyperbola; and, if the sign of a be different from
TO FIND THE FOCI. • 207
that of 7, the real foci will lie on the axis of x. Hence to find the eccentricity with reference to a real focus ; obtain the values of a and ^ from (iii) and (iv), then (ii) will give the eccentricity required, if we take for a that value whose sign is different from the sign of 7.
Ex. Find the eccentricity of the conic whose equation is a;2 - ixy ~2tf+10x + 4:y=0.
The equation referred to the centre is x^ - Axy - 2?/^ -1 = 0. This will become ax^ + ^y^- 1 = 0, where a + ^=-l and a^ = -6. Hence a=2. /3 = -3. The eccentricity with reference to a real focus is given by 2 = - 3 (1 - e"^) ; therefore e = V|.
192. To find the foci of a conic.
Let (x\ y) be one of the foci of the conic
<2^' + 2 A^y + %' + 2^^' + 2/2/ + c = 0 (i) .
The corresponding directrix of the conic is the polar of {x , y)\ therefore its equation is
X {ax +hy+g)-\ry Qix' + hy +/) + gx' +fy + c = 0. The equation of the conic may therefore be written in the form {x - xj + (2/ - yj -\[x {ax' + hy' +g)-\-y (hx' + hf -!-/)
+9^'+fl/' + oY = 0 (ii).
Since (i) and (ii) represent the same curve, the coeffi- cients in (ii) must be equal to the corresponding coefficients in (i) multiplied by some constant. We have therefore 1 - \ (ax' + hy' + gy = ka,
-\(ax+hy+g) Qix +hy •\-f) =^kh,
1 - X {hx' + hy' +fy = kb,
-x'-\{ax'-\-hy' + g) (go)' -h fy' + c) =kg,
-y'-X{hx' + hy'+f) {gx+fy' + c) ==kf
and x"^ + y"" - "^' (g^' +fy + cY — kc.
From the first three of the above equations we have {ax' + hy'+gY-jhx' + hy'+P) a-b
_ (ax' + hy'+g){hx' + hy'-\'f) . h '"^ ^'
208 TO FIND THE FOCI.
Multiply the fourth and fifth equations by x , y respec- tively and add them to the sixth ; then, comparing with the second, after rejecting the factor gx -\-fy + c, we get
X {ax + %' + g)+y' (/'^' + hy +/) + gx +fy' + c
= (gg" + h\j + g) {hx + by + f)
, . , ,. {ax + hy + g) {hx + by +/) ,. , or <l>{x, y) = ^ ^~~h ' ' ' ^^^)*
The four foci are therefore from (iii) and (iv) the four points of intersection of the two conies
{ax + hy + gY - {hx + by +fy ^ {ax + hy +g) {hx + by +/) a — b h
= 4> (^» y)'
193. The equation of a conic referred to a focus as origin is x^ + y'^ = ^ {x cos a + y sina — 'pf.
Either of the lines x + J— 1 y = 0 meets the conic in coincident points.
Hence the tangents from the focus to the conic are the imaginary lines x ±y J— 1 = 0, or as one equation
Since the equation of the tangents from a focus is in- dependent of the position of the directrix, it follows that if conies have one focus common they have two imaginary tangents common, and that confocal conies have four common tangents.
Now if the origin and axes of co-ordinates be changed in any manner, the equation of the tangents from a focus will be changed from
x'i-y^ = 0iox^ + y'-]-2gx + 2fy-hc = 0.
Hence the equation of the tangents to a conic from a focus satisfies the conditions for a circle.
We may therefore find the foci of a conic in the following manner.
THE AXES.
The equation of the tangents from {x, y) to the conic <f> {x, 2/) = 0 is {ax^ + 2hxy + hy'' + 2y^ + ^fy + c) </> {x, y)
= {axx + h {xy + /^) + hy'y + y (a? + ^') 4-/(2/ + y') + c}l If {x , y) be a focus of the conic, this equation satisfies the conditions for a circle, viz. that the coefficients of x^ and y^ are equal, and that the coefficient of xy is zero. Hence we have «</> (a;; 2/0 - {ax' + %' +^)^ = h^ {x\ y') - Qix' + hy' -^/)^ and /i0 {x, y') = (««' + hy + ^) (/i^' + hy' -f /).
The foci are therefore the points given by {ax + % + gr)'^ - (/la; + 62/ +/)' a — 6
^ (ax + hy+ff) (hx + hy +/) ^ ^ ^^^ ^^^
/Z/
The equations giving the foci may be written \c?^/ \c?y/ dx dy . .
— ^Tir-=-r-=**-
194. To find the equation of the axes of a conic.
The axes of a conic bisect the angles between the asymptotes, and the asymptotes are parallel to the lines given by the equation ax^+ 2hyx + hy^=0 [Art. 174]. Hence [Art. 39] the axes are the straight lines through the centre of the conic parallel to the lines given by the equation
a — b h '
We may also find the equation of the axes as follows.
If a point Pbe on an axis of the conic, the line joining P to the centre of the conic is perpendicular to the polar of P.
Let x\ y be the co-ordinates of P, then the equation of the polar of P is
x{ax' + hy + g) â– \-y{hx -^hy' +f) -^ gx +fy -{â– c = 0...{i).
S. C. s. 14
210 THE AXES."
The equation of any line through the centre of the
conic is ax-\-hy + g -\-Xgix-\-hy +f) = 0 (ii).
Since (ii) is perpendicular to (i), we have (a+X/fc) {ax+hy-\-g) + {h-{-\h) {hx+by-^f)=0. . . (iii). Since (ii) passes through {x\ y'), we have
ax + hy' +g + \ {hx + by +/) = 0 (iv).
Eliminate X from (iii) and (iv), and we see that {x\ y') must be on the conic {ax â– \-hy-\- gf- {hx + hy +/)'' _ [ax + hy + g) {hx -\-hy-{-f)
a-b h
which is the equation required.
The equation of the axes may also be deduced from Aiticle 192 or 193; for one of the conies on which we have found that the foci lie passes through the centre, and therefore must be the axes.
Ex. 1. Shew that all conies through the four foci of a conic are rectangular hyperbolas.
Ex. 2. Prove that the foci of the conic whose equation is
ax^ + 2hxy + by- = 1, lie on the curves
7? - y^ _xy 1
a-h ~ h ~ h^-ab ' Ex. 3. Shew that the real foci of the conic
x^-6xy + y^-2x-2y + 5=^0 Are (1, 1) and (- 2, -2). Ex. 4. The co-ordinates of the real foci of 2x^ - 8xy - 4y^ -4y + l = 0
(o.^)and(-|,-|).
Ex. 5. The focus of the parabola x^ + 2xy + y--4M + Sy -(j=0 is the point (-^, -1).
Ex. 6. Shew that the product of the perpendiculars from the two imaginary foci of an ellipse on any tangent to the curve is equal to the square of the semi-major axis.
Ex. 7. Shew that the foot of the perpendicular from an imaginary focus of an ellipse on the tangent at any point lies on the circle described on the minor axis as diameter,
I
TANGENT AND NORMAL AS AXES. 211
Ex. 8. If a circle have double contact with an ellipse, shew that the tangent to the circle from any point on the ellipse varies as the distance of that point from the chord of contact.
195. To find the equation of a conic when the aooes of co-ordinates ai-e the tangent and normal at any point.
The most general form of the equation of a conic is ax^ 4- '^hxy + by"" + ^gx + 2/y + c = 0.
Since the origin is on the curve, the co-ordinates (0, 0) will satisfy the equation, and therefore c = 0.
The line y = 0 meets the curve where ax^ + 2gx = 0. If y = 0 is the tangent at the origin, both the values of x given by the equation ax^ + ^gx — 0 must be zero ; there- fore ^ = 0.
Hence the most general form of the equation of a conic, when referred to a tangent and the corresponding normal as axes of x and y respectively, is
aa?^ + ^hxy + hy^ + 2/^/ = 0.
Ex. 1. All citords of a conic which subtend a right angle at a fixed point O on the conic, cut the normal at O in a fixed point.
Take the tangent and normal at 0 for axes ; then the equation of the conic will be
ax^ + 2hxy + hy"^ + 2fy = 0.
Let the equation of PQ, one of the chords, be Ix + my -1 = 0. The equation of the lines OP, OQ will be [Art. 38]
ax^ + 2hxy + bj/^ + 2fy{lx + mi/)=0 (i).
But OP, OQ are at right angles to one another, therefore the sum of the coefficients of ar* and y^ in (i) is zero. Hence We have a + b + '2fm=0 ; which shews that vi is constant, and m is the reciprocal of the intercept on the normal.
Ex. 2. If any tico chords OP, OQ of a conic make equal angles with the tangent at O, the line PQ will cut that tangent in a fixed point.
196. The equation of the normal at any point {x\ y') of the conic whose equation is ax^ + by^ = 1 is
X — X _ y — y ax by
14—2
212 NORMALS.
This will pass through the point (A, h) if h — x k — y' ax' ~~ bi/' ' i. e. if x'y' (a — b)+ hhy — akx — 0.
Therefore the feet of the normals which pass through a particular point (/i, U) are on the conic
xy {a — h) +hhy — akx = 0 (i).
The four real or imaginary points of intersection of the conic (i) and the original conic are the points the normals at which pass through the point (h, k).
The conic (i) is clearly a rectangular hyperbola whose asymptotes are parallel to the axes of co-ordinates, that is to the axes of the original conic. It also passes through the centre of that conic, and through the point {h^ k) itself
197. If the normals at the extremities of the two chords Ix + my — 1 = 0 and Fx + m'y — 1 = 0 meet in the point (h, k), then, for some value of \, the conic
ax' + by^-l-X {Ix + my-l) {I'x + my - 1) = 0. ..(i), which, for all values of \, passes through the four extremities of the two chords, will [Art. 196] be the same as
xy {a—b) + hhy — akx — 0 (ii).
The coefficients of x^ and y^, and the constant term are all zero in this last equation, and therefore they must be zero in the preceding.
We have therefore
a-\ir==0, b-\mm=0, and 1 + X = 0.
Hence, if the normals at the ends of the chords Ix + Qiiy — 1 = 0 and Ix + my — 1=0 meet in a point, we have
U mm' ^ .....
- = -^- = -1 (ill)-
a 0
198. By the preceding Article we see that normals to the ellipse whose axes are 2a, 2h at the extremities of the
NORMAL!^. 213
chords whose equations are
Ix -f- my — 1 = 0, and I'x + my —1=0,
â– will meet in a point, if
aHl' = }fmm=^-l (i).
If the eccentric angles of these four points be a, /5 and 7, 3, the equations of the chords will be
X « + /?.'?/. OL + I^ OL — P
, X y+^ , y . y+^ y—B
and - cos ^ h f sm ^ = cos ^— .
a 2 0 2 2
We have therefore, by comparing with (i),
a + /S 7 + S , a-13 y-S _ cos — ^ cos -^ — h cos —J, cos ^-^— = 0,
, . . a + /3 . 7+S a-yS y-S „
and sm — ~ — sm ^-^ — f- cos — - — cos -~- = 0. Zi Ji li JL
By subtraction, we have
cos -.-C1__A„ = 0,
whence « +^ _]. ry _}. g = (2/z + 1) tt (ii).
Also the first equation gives
cos ^ + cos "—^ + cos ^' ^- —
+ cos ~^-—^- = ^. and, using the condition (ii), this becomes
sin (a + /3) + sin (/9 + 7) + sin (7 + a) = 0. . .(iii).
Ex. 1. If ABC be a maximum triangle inscribed in an ellipse, tliA normals at A, B, G will meet in a point.
The eccentric angles will be a, a + — , and a + -^[Art. 138]. Tlio
O O
214 NORMALS.
condition that the normals meet in a point is [Art. 198 (iii)]
sin2a + sinf 2a + -^ j+8in(2a + -^ J— 0, which is clearly true.
Ex. 2. The nwmals to a central conic at the four points P, Q, R, S vieet in a point, and the circle through P, Q, R cuts the conic again in S'; shew that SS' is a diameter of the conic.
SS' will be a diameter of the conic if RS and RS' are parallel to conjugate diameters [Art. 134].
Now if PQ be Ix + my -1 = 0, RS will hejx+-y + l = 0 [Art. 197]; also RS' will be parallel to lx-my = 0, since P, Q, R, S' are on a circle; hence SS' is a diameter, for [Art. 132] lx-iny = 0, and jx+ -y = 0 are conjugate diameters of ax^ + by^=l.
[The proposition may also be obtained from Art. 198 (ii), and Art. 186, Ex. (1).]
Ex. 3. If the normals to an ellipse at A, B, C, D meet in a point, tlie axis of a parabola through A, B, G, D is parallel to one or otiier of the equi-conjugates.
If h, k be the point where the normals meet, A, B, C, D are the four points of intersection of the conies
All conies through the intersections are included in the equation
If this be a parabola the terms of the second degree must be a perfect square, and therefore niust be the square of - ± ^ . The equation of every
such parabola is therefore of the form ( - ± | J +Ax + By+C = 0. Their
axes are therefore [Art. 172] parallel to one or other of the lines "^ ± - =0.
a b
Ex. 4. Tlie perpendicular from, any point P on its polar with respect to
a conic passes through a fixed point 0; prove (a) that the locus of P is a
rectangular hyperbola, (^) that the circle circumscribing the triangle which
the polar of P cuts off from the axes always passes through a fixed point 0',
(7) that a parabola lohose focus is 0' tvill touch the axes and all such
polars, (5) that the directrix of this parabola is CO, where G is the centre
of the conic, and (e) that O and 0' are interchangeable.
NOKMALS. 215
Let the equation of the conic be -g + i^ = ^-> and let {h, k) be the co-or- dinates of the fixed point 0.
If the co-ordinates of any point P be (.r', y'), the equation of the line through P perpendicular to its polar with respect to the conic will be x-x' y-y'
a^x hhi „ -„ or —T r = « -^•
If this line pass through the point (/i, k), we have
___=„- 6- w.
From (i) we see that (x', y') is on a rectangular hyperbola (a).
The equation of the circle circumscribing the triangle cut off from the axes by the polar of {x\ y') will be
a^x h^y
« y
The circle will pass through the point {\h, - \k) if
x' y' Hence, if {x', y') satisfies the relation (i), we have
Hence the circles all pass through the point 0' whose co-ordinates are
FTP'Sl^T/c-^^ <^^-
The point O' is on the circle circumscribing the triangle formed by the axes and any one of the polars ; hence the parabola whose focus is 0' and which touches the axes will touch every one of the polars (7).
The parabola touches the axes of the original conic, therefore the centre C is a point on the directrix of the parabola. Also the lines CO and CO' make equal angles with the axis of x, which is a tangent to the parabola; therefore 0' being the focus, CO is the directrix (5).
Since CO' . CO = a^-b^, and CO, CO' make equal angles with the axis of X, and are on the same side of the axis of y, the points 0 and 0' are interchangeable (e).
199. Definition. Two curves are said to be similar and sirnilarly situated when radii vectores drawn to the first from a certain point 0 are in a constant ratio to
216 SIMILAR CONICS.
parallel radii vectores drawn to the second from another point 0\
Two curves are similar when radii drawn from two fixed points 0 and 0' making a constant angle with one another are proportional.
The two fixed points 0 and 0' may be called centres of similaHty.
200. If one pair of centres of similarity exist for two curves, then there will be an infinite number of such pairs.
Let 0, 0' be the given centres of similarity, and let OP, OF' be any pair of parallel radii. Take G any point whatever, and draw O'G parallel to OG and in the ratio O'F' : OP. Then, from the similar triangles GOP, and G'O'P' we see that GP is parallel to G'F and in a constant ratio to it ; which proves that G, G' are centres of similarity.
201. If two central conies he similar the centres of the two curves will be centres of similarity.
Let 0 and 0' be two centres of similarity. Draw any chord POQ of the one, and the corresponding chord FO' (3' of the other. Then by supposition PO. OQ.P'O'. O Q' is constant for every pair of corresponding chords. But since 0 is a fixed point PO . OQ is always in a constant ratio to the square of the diameter of the first conic which is parallel to it. The same applies to the other conic. Therefore corresponding diameters of the two conies are in a constant ratio to one another; this shews that the centres of the curves are centres of similarity.
202. To find the conditions that two conies may be similar and similarly situated.
By the preceding Article, their respective centres are centres of similarity.
Let the equations of the conies referred to those centres and parallel axes be
ax"^ + 2hxy + by^ + c =0, and a'sif + 2h'xy + by + c' = 0 :
SIMILAR CONICS. 217
or, in polar co-ordinates,
r' (acos' d+2h sin (9 cos <9 + 6 sin' d) + c = 0, and ?•'' [a cos' d + 2Ji sin ^ cos ^ + b' sin^ 6} -\- c = 0. If therefore r' : ?*"^ be constant, we must have a cos'^ + 2h sin ^ cos ^ + 6 sin'^^ a' cos""^ + 2N sin ^ cos ^ + h' sin'' 6/ the same for all values of 6.
This requires that ~7= p — ii - Hence the asymptotes
of the two conies are parallel. [This result may be obtained in the following manner : since r : / is constant, when one of the two becomes infinite, the other will also be infinite, which shews that the asymptotes are parallel.]
Conversely, if these conditions be satisfied, and if each fraction be equal to X, then
r"^ _ c
therefore the ratio of corresponding radii is constant, and therefore the curves are similar.
If c and Xc' have not the same sign the constant ratio is iinaginary, and is zero or infinite if c or c' be zero.
The conditions of similarity are satisfied by the three curves whose equations are
xy = c,xy = 0, and xy== — c. Therefore an hyperbola, the conjugate hyperbola, and their asymptotes are three similar and similarly situated curves ; the constant ratio being ^/ — 1 for the conjugate hyperbola, and zero for the straight lines.
These curves have not however the same shape. For similar curves to have the same shape the constant ratio must be real and finite.
203. To find the condition that two conies may he similar although not similarly situated.
We have seen that the centres of the two curves must be centres of similarity.
218 SIMILAR CONICS.
Let the equations of the curves referred to their respective centres be
ax^-\-2hxij +hy'''\-c =0 (i),
aV + ^h'xy + h'y^ + c' = 0 (ii),
and let the chord which makes an angle 6 with the axis of X in the first be proportional, for all values of 6, to that which makes an angle {6 + a) in the second. If the axes of the second conic be turned through the angle a, we shall then have radii of the two conies which make the same angle with the respective axes in a constant ratio.
Let the equation of the second conic become
:4 V + ^H'xy + 5y + c = 0.
Then, by the preceding Article, we must have
a h ~ b'
therefore A' + F _^(A^B' - H'^)
theretore a+2> " ^{ab-JiT'
But [Art.52] A'f B = a'-v h', ^ndA'B'-H''= a'h'- U' ; therefore the condition of similarity is
ah-Pa'b'-h'' (a+by~{a'-\-b'y'
The above shews that the angles between the asymp- totes of similar conies are equal. [See Art. 174.]
This result may also be obtained in the following manner : since radii vectores of the two curves which are inclined to one another at a certain constant angle are in a constant ratio, it follows that the angle between the two directions which give infinite values for the one curve must be equal to the corresponding angle for the other, that is to say the angle between the asymptotes of the one conic is equal to the angle between the asymptotes of the other.
EXAMPLES ON CHAPTER X. 219
Examples on Chapter X.
1. If ^ and P be any two points, and C the centre of a ct)nic; shew that the perpendiculars from Q and G on the polar of P with respect to the conic, are to one another in the same ratio as the perpendiculars from P and G on the polar of Q.
2. Two tangents drawn to a conic from any point are in the same ratio as the corresponding normals.
3. Find the loci of the fixed points of the examples in Article 195, for different positions of 0 on the conic.
4. POQ is one of a system of parallel chords of an ellipse, and 0 is the point on it such that PO^ + OQ^ is constant; shew that, for different positions of the chord, the locus of 0 is a concentric conic.
5. If 0 be any fixed point and OPP any chord cutting a conic in P, P\ and on this line a point D be taken such that
777)2 — Typi "^ Tyivs' ^^® locus of D will be a conic whose centre
is 0.
G. If OPPQQ' is one of a system of parallel straight lines cutting one given conic in P, P' and another in Q, Q\ and 0 is such that the ratio of the rectangles OP. OP' and OQ . OQ' is constant; shew that the locus of 0 is a conic through the inter- sections of the original conies.
7. POP, QOQ' are any two chords of a conic at right angles to one another through a fixed point 0 ; shew that
8. If a point be taken on the axis-major of an ellipse, whose abscissa is equal to a ^1 -^ — ,, prove that the sum of
the squares of the reciprocals of the segments of any chord passing through that point is constant.
220 EXAMPLES ON CHAPTER X.
9. If PP' be any one of a system of parallel chords of a rectangular hyperbola, and Ay A' be the extremities of the perpendicular diameter; PA and P'A' will meet on a fixed circle. Shew also that the words rectangular hyperbola, and circle, can be interchanged,
10. If PSP' be any focal chord of a parabola and PJ/, P M' be perpendiculars on a fixed straight line, then will
PM P'M'
PS "^ m
be constant.
11. Chords of a circle are drawn through a fixed point and circles are described on them as diameters; prove that the polar of the point with regard to any one of these circles touches a fixed parabola.
1 2. From a fixed point on a conic chords are drawn making equal intercepts, measured from the centre, on a fixed diameter ; find the locus of the point of intei-section of the tangents at their other extremities.
13. If {x\ y) and {x"^ y") be the co-ordinates of the extremities of any focal chord of an ellipse, and £c, y be the co-ordinates of the middle point of the chord ; shew that y' y" will vary as x. What does this become for a parabola %
14. Sy U are two fixed points on the axis of an ellipse equidistant from the centre C; PSQ^ PHQ' are chords through them, and the ordinate MQ is produced to ^ so that MR may be equal to the abscissa of Q' ; shew that the locus of R is a rectangular hyperbola.
15. S, II are two fixed points on tlie axis of an ellipse equidistant from the centre, and PSQ^ PHQ' are two chords of the ellipse; shew that the tangent at P and the line QQ make angles with the axis whose tangents are in a constant ratio.
16. Two parallel chords of an ellipse, drawn through the
foci, intersect the curve in points P, P' on the same side of the
major axis, and the line through P, P' intersects the semi-axes
AC^ BC* CA,CPin C/,F respectively: prove that 71772+ yr^ ^^ invariable.
EXAMPLES ON CHAPTER X. 221
1 7. From an external point two tangents are drawn to an ellipse ; shew that if the four points where the tangents cut the axes lie on a circle, the point from which the tangents are drawn will lie on a fixed rectangular hyperbola.
18. Prove that the locus of the intersection of tangents to an ellipse which make equal angles with the major and minor axes respectively, but which are not at right angles, is a rect- angular hyperbola whose vertices are the foci of the ellipse.
19. If a pair of tangents to a conic meet a fixed diameter in two points such that the sum of their distances from the centre is constant; shew that the locus of the point of intersec- tion is a conic. Shew also that the locus of the point of inter- section is a conic if the product, or if the sum of the reciprocals be constant.
20. Through 0, the middle point of a chord AB of an ellipse, is drawn any chord POQ. The tangents at P and Q meet AB in S and T respectively. Prove that AS = BT.
21. Pairs of tangents are drawn to the conic aoif ■¥ pif ~ 1 so as to be always parallel to conjugate diameters of the conic aa? + 2hxi/ + by^ = 1 ; shew that the locus of their intersection is
ax' + h'lf + 2hxy = - + — .
22. FTf FT' are two tangents to an ellipse which meet the tangent at a fixed point Q in T^ T ; find the locus of P (i) when the sum of the squares of QT and QT' is constant, and (ii) when the rectangle QT . QT' is constant.
23. 0 is a fixed point on the tangent at the vertex ^ of a conic, and P, P', are points on that tangent equally distant from 0; shew that the locus of the point of intersection of the other tangents from P and P' is a straight line.
24. If from any point of the circle circumscribing a given square tangents be drawn to the circle inscribed in the same square, these tangents will meet the diagonals of the square in four points lying on a rectangular hyperbola.
222 EXAMPLES ON CHAPTEli X.
25. Find the locus of the point of intersection of two tangents to a conic which intercept a constant length on a fixed straight line.
26. Two tangents to a conic meet a fixed straight line MX in F, Q: ii P, Q he such that FQ subtends a right angle at a fixed point 0, prove that the locus of the point of inter- section of the tangents will be another conic.
27. The extremities of the diameter of a circle are joined to any point, and from that point two tangents are drawn to the circle; shew that the intercept on the perpendicular diameter between one line and one tangent is equal to that between the other line and the other tangent.
28. Triangles are described about an ellipse on a given base which touches the ellipse at P; if the base angles be equidistant from the centre, prove that the locus of their vertices is the normal at the other end of the diameter through P.
29. A parabola slides between rectangular axes; find the curve traced out by any point in its axis; and hence shew that the focus and vertex will describe curves of which the equations are ocy = a' {a^ + 2/'), afy' {x' + f + 3a') = a%
4:a being the latus rectum of the parabola.
30. If the axes of co-ordinates be inclined to one another at an angle a, and an ellipse slide between them, shew that the equation of the locus of the centre is
sin'a (x' -Vlf - l^Y ~ ^ cos^a (oi^'if sin^a - q^) = 0,
where j^'' and q^ denote respectively the sum of the squares and the product of the semi-axes of the ellipse.
31. If OP, OQ are two tangents to an ellipse, and CP', CQ' the parallel semi-diameters, shew that
OF.OQ + CF . CQ' = OS. OH,
S, H being the foci.
32. Through two fixed points P, Q straight lines A FB, CQD are drawn at right angles to one another, to meet one given straight line \n A^ G and another given straight line parpen-
EXAMPLES ON CHAPTER X. 223
diculav to the former in B^ D ; find the locus of the point of intersection of AD^ BC ; and shew that, if the line joining P and Q subtend a i-ight angle at the point of intersection of the given lines, the locus will be a rectangular hyperbola.
33. Prove that the locus of the foot of the perpendicular from a point on its polar with respect to an ellipse is a rect- angular hyperbola, if the point lies on a fixed diameter of the ellipse.
34. The polars of a point P with respect to two concentric and co-axial conies intersect in a point Q; shew that if P moves on a fixed straight line, Q will describe a rectangular hyperbola.
35. Shew that if the polars of a point with respect to two given conies are either parallel or at right angles the locus of the point is a conic.
36. The line joining two points A and B meets the two lines OQ, OP in ^andP. A conic is described so that OP and OQ are the polars of A and B with respect to it. Shew that the locus of its centre is the line OR where R divides AB so that
AR : RB :: QR : RP.
37. Find the locus of the foci of all conies which have a common director-circle and one common point.
38. Shew that the locus of the foci of conies which have a given centre and touch two given straight lines is an hyperbola.
39. In the conic ax^+ 2hxy + hif = 2y, the rectangle under the focal distances of the origin is
1 ah - U '
40. The focus of a conic is given, and the tangent at a given point; shew that the locus of the extremities of the conjugate diameter is a parabola of which the given focus is focus.
41. A series of conies have their foci on two adjacent sides of a given parallelogram and touch the other two sides; shew that their centres lie on a straight line,
224 EXAMPLES ON CHAPTER X.
42. If TP, TQ be tangents drawn from any point T to touch a conic in P and Q, and if S and H be the foci, then
SPTSQ~ ifP . HQ'
43. An ellipse is described concentric with and toiiching a given ellipse and passing through its foci ; shew that the locus of the foci of the variable ellipse is a lemuiscate.
44. Having given five points on a circle of radius a; shew that the centres of the five rectangular hyperbolas, each of which passes through four of the points, will all lie on a circle of
radius ^ .
45. Tf a rectangular hyperbola have its asymptotes parallel to the axes of a conic, the centre of mean position of the four points of intersection is midway between the centres of the curves.
46. Three straight lines are drawn parallel respectively to tlie three sides of a triangle ; shew that the six points in which they cut the sides lie on a conic.
47. If the normal at P to an ellipse meet the axes in the
2 11
points Gy G'j and 0 be a point on it such that ^^ = ^^ + -j-^ ;
X (/ P(jr rix
then will any chord through 0 subtend a right angle at P.
48. Through a fixed point 0 of an ellipse two chords OP, OP' are drawn; shew that, if the tangent at the other extremity 0' of the diameter through 0 cut these lines pro- duced in two points Q, Q' such that the rectangle O'Q . O'Q' is constant, the line PP' will cut 00' in a fixed point.
49. A chord LM is drawn parallel to the tangent at any point P of a conic, and the line PR which bisects the angle LPM meets LMin P; prove that the locus of ^ is a hyperbola having its asymptotes parallel to the axes of the original conic.
50. A given centric conic is touched at the ends of a chord, drawn through a given point in its transverse axis, by another conic which passes through the centre of the former : ])i ove that the locus of the centre of the latter conic is also a centric conic.
EXAMPLES ON CHAPTER X. 225
51. QQ' is a chord of an ellipse parallel to one of the equi- conjugate diameters, C being the centre of the ellipse; shew that the locus of the centre of the circle QCQ' for different positions of QQ' is an hyperbola.
52. A circle is drawn touching the ellipse — + f^=l at
ah
any point and passing through the centre; shew that the locus of the foot of the perpendicular from the centre of the ellipse on the chord of intersection of the ellipse and circle is the
ellipse aV + hY = 7^2-_ ^2x2 •
53. Find the value of c in order that the hyperbola
of if 2xy — c = 0 may touch the ellipse -2 + fa — 1 = 0, and shew that
the point of contact will be at an extremity of one of the equi-conjugate diameters of the ellipse.
Shew also that the polars of any point with respect to the two curves will meet on that diameter.'
54. Shew that, if CD, EF be parallel chords of two circles which intersect in A and B, a conic section can be drawn through the six points A, B, C, D, E, F; and give a construc- tion for the position of the major axis.
55. If the intersection P of the tangents to a conic at two of the points of its intersection with a circle lie on the circle, then the intersection P' of the tangents at the other two points will lie on the same circle. In this case find the relations con- necting the positions of P and P' for a central conic, and deduce the relative positions of P and P' when the conic is a parabola.
56. If Ty T be any two points equidistant and on opposite sides of the directrix of a parabola, and TP^ TQ be the tangents to the parabola from T^ and T'Q\ T'F the tangents from T ; then will T^ P, Q, T\ P\ Q' all lie on a rect- angular hyperbola.
57. If a straight line cut two circles in ^, A' and B^ B' and if C, C be the common points of the circles, and if 0 be any point; shew that the three circles OAA' , OBB' and OCC will have a common radical axis.
S. C. S. 15
226 EXAMPLES ON CHAPTER X.
58. With a fixed point 0 for centre circles are described cutting a conic ; shew that the locus of the middle points of the common chords of a circle and of the conic is a rectangular hyperbola.
59. With a fixed point 0 for centre any circle is described cutting a conic in four points real or imaginary; shew that the locus of the centres of all conies through these four points is a rectangular hyperbola, which is independent of the radius of the circle.
60. The normals at the ends of a focal chord of a conic intersect in 0 and the tangents in T \ shew that TO produced will pass through the other focus.
61. If from any point four normals be drawn to an ellipse meeting an axis in G^^ G^, G^, G^, then will
1111 4
+ 7T7T + 777V
CG^ CG^ CG^ CG^ CG^ + CG^+CG^ + CG/
62. If the normals to an ellipse at ^, B, C, J) meet in 0, find the equation of the conic ABC DO, and shew that the locus of the centre of this conic for a fixed point 0 is a straight line if the ellipse be one of a set of co-axial ellipses.
63. The four normals to an ellipse at P, Q, i?, S meet at 0. Straight lines are drawn from F, Q, R, S such that they make the same angles with the axis of the ellipse as CP, CQ, Ci?, CS respectively : prove that these four lines meet in a point.
64. The normals at P, Qy B, S meet in a point 0 and lines are drawn through F, Q, E, S making with the axis of the ellipse the same angles as OF, OQ, OB, OS respectively : prove that these four lines meet in a point.
65. The normals at F, Q, B, S meet in a point ; and F', Q', B', *S" are the points of the auxiliary circle correspond- ing to F, Q, B, S respectively. If lines be drawn through F, Q, B, S parallel to FC, Q'Q, B'G and S'C respectively, shew that they will meet in a point.
66. If from a vertex of a conic perpendiculars be drawn to the four normals which meet in any point 0, these lines will meet the conic again in four points on a circle.
EXAMPLES ON CHAPTER X, 227
67. Tangents are drawn from any point on the conic ~a + Ta = ^ to the conic -„ + 75 = 1 : prove that the normals at
the points of contact meet on the conic a^x^-\- h^y^ — ( — - — j .
68. liABC be a triangle inscribed in an ellipse such that the tangents at the angular points are parallel to the opposite sides, shew that the normals a,t A, B, C will meet in some point 0. Shew also that for different positions of the triangle the locus of 0 will be the ellipse 4aV + iby = {a' - by.
69. If the co-ordinates of the feet of the normals to xi/ = a from the point (X,Y) be x„ y^', x^, y^; x^, y^; x^, y^) then ^" = 2/, + 2/a + 2/3 + ^4' ^^^ X^x^^ x^ + iC, + x^.
70. The locus of the point of intersection of the normals to a conic at the extremities of a chord which is parallel to a given straight line, is a conic.
71. Any tangent to the hyperbola 4^xy = ab meets the
ellipse —2 + 0=1 in points P, Q ; shew that the normals to the
Gi 0
ellipse at P and Q meet on a fixed diameter of the ellipse.
72. If four normals be drawn from the point 0 to the ellipse Ifx^ + ay = a^6^, and p^, p^, p^, p^ be the perpendiculars from the centre on the tangents to the ellipse drawn at the feet of these normals, then if
JL J_ 1 Jl-I P.'^P.'''Ps'''prc'' where c is a constant, the locus of 0 is a hyperbola.
73. Find the locus of a point when the sum of the squares of the four normals from it to an ellipse is constant.
74. The tangents to an ellipse at the feet of the normals which meet in (/,^) form a quadrilateral such that if {x',y), {od\y")
be any pair of opposite vertices — ^ = ^- = — 1, and that the
equation of the line joining the middle points of the diagonals of the quadrilateral is ^ + ^y = 0.
75. Tangents are drawn to an ellipse at four points which are such that the normals at those points co-intersect; and four rectangles are constructed each having two adjacent sides along
15-2
228 EXAMPLES ON CHAPTER X.
the axes of the ellipse, and one of those tangents for a diagonal. Prove that the distant extremities of the other diagonals lie in one straight line.
76. From a point P normals are drawn to an ellipse meeting it in ^, B, C, D. If a conic can be described passing through A^ B, C, D and a focus of the ellipse and touching the corresponding directrix, shew that P lies on one of two fixed straight lines.
77. If the normals 2X A, B^ Gy B meet in a point 0, then will SA,SB,SC.SD = k\ S0\ where aS' is a focus.
78. From any point four normals are drawn to a rect- angular hyperbola; prove that the sum of the squares on these normals is equal to three times the square of the distance of the point from the centre of the hyperbola.
79. A chord is drawn to the ellipse -2 + ^^=1 meeting the major axis in a point whose distance from the centre is
- — —. At the extremities of this chord normals are drawn a + 6
to the ellipse; prove that the locus of their point of intersection
is a circle.
80. The product of the four normals drawn to a conic from any point is equal to the continued product of the two tangents drawn from that point and of the distances of the point from the asymptotes.
81. Find the equation of the conic to which the straight lines (x + Xyf -p^ = 0, and (x + /xyY -q^=0 are tangents at the ends of conjugate diameters.
82. From any point T on the circle of + y^ = c^, tangents
TP, TQ are drawn to the ellipse -^ + ^= 1, and the circle TPQ
cuts the ellipse again in P'Q'. Shew that the line P'Q' always touches the ellipse
x' y' _ c'
83. A focal chord of a conic cuts the tangents at the ends of the major axis vn. A, B; shew that the circle on AB as diameter has double contact with the conic:
EXAMPLES ON CHAPTER X.
84. ABCD is any rectangle circumscribing an ellipse whose foci are S and H; shew that the circle ABS or ABH is equal to the auxiliary circle.
85. Any circle is described having its centre on the tangent at the vertex of a parabola, and the four common tangents of the circle and the parabola are drawn; shew that the sum of the tangents of the angles these lines make with the axis of the parabola is zero.
86. Tangents to an ellipse are drawn from any point on the auxiliary circle and intersect the directrix in four points : prove that two of these lie on a straight line passing through the centre, and find where the line through the other two points cuts the major axis.
87. If w =- 0, i; = 0 be the equation of two central conies, and u^, v^ the values of u, v at the centres (7, C of these conies respectively, shew that u^v = v^u is the equation of the locus of the intersection of the lines CP, G'P\ where P, P' are two points, one on each curve, such that PP' is parallel to CC\ Examine the case where the conies are similar and similarly situated.
88. Two circles have double internal contact with an ellipse and a third circle passes through the four points of contact. If ^, t', T be the tangents drawn from any point on the ellipse to these three circles, prove that tt' —T^.
89. Find the general equation of a conic which has double contact with the two circles {x-af-vy^=c^^ {x—l)f-\-y^ = (f, and prove that the equation of the locus of the extremity of the latus rectum of a conic which has double contact with the circles {x^aY + y'^ c" is if {of - a') (af-a' + c') = c V.
90. Shew that the lines Ix + my = 1 and I'x + m'y = 1 are conjugate diameters of any conic through the intersections of the two conies whose equations are
(Z^m' - I'^m) a? -\-2{l-l') jnm'xy + (m - m') mm'f = 2 {Im' - I'm) x^ and imH' - m'H) y^ + 2{m- m') ll'xy + {I - V) IV t? = 2 {inl! - m'l) y.
91. If through a fixed point chords of an ellipse be drawn, and on these as diameters circles be described, prove that the other chord of intersection of these circles with the ellipse also passes through a fixed point.
230 EXAMPLES ON CHAPTER X.
92. The angular points of a triangle are joined to two fixed points ; sliew that the six points, in which the joining lines meet the opposite sides of the triangle, lie on a conic.
93. If three sides of a quadrilateral inscribed in a conic pass through three fixed points in the same straight line, shew that the fourth side will also pass through a fixed point in that straight line.
94. Two chords of a conic PQ, P'Q' intersect in a fixed point, and PP passes through another fixed point. Shew that QQ' also passes through a fixed point, and that PQ\ P'Q touch a conic having double contact with the given conic.
95. A line parallel to one of the equi-conjugate diameters of an ellipse cuts the tangents at the ends of the major axis in the points P, Q, and the other tangents from P, Q to the ellipse meet in 0 ; shew that the locus of 0 is a rectangular hyperbola.
96. Z, Jf, Ny R are fixed points on a rectangular hyper- bola and P any other point on it, PA is perpendicular to LM and meets NR in a, PC is perpendicular to LN and meets MR in c, PB is perpendicular to LR and meets MN in h. Prove that PA . Pa = PB.Ph^PC.PG.
97. P is any point on a fixed diameter of a parabola. The normals from P meet the curve in ^, B^ C. The tangents parallel to PA, PB, PC intersect in A', B\ C . Shew that the ratio of the areas of the triangles ABC, AB'C is constant.
98. A point P is taken on the diameter AB oi 2, circle whose centre is C. On AP, BP as diameters circles are described : the locus of the centre of a circle which touches these three circles is an ellipse having C for one of its foci.
99. The straight lines from the centre and foci S, S' of a conic to any point intersect the corresponding chord of contact in F, G, C ; prove that the radical axis of the circles described on SG, S'G' as diameters passes through F.
100. If the sides of a triangle ABC meet two given straight lines in a^, a^; b^, h^; Cj, c^ respectively; and if round the quadrilaterals hfi^c^c^, ^i<^3*i^2» ^i^a^fia conies be de- scribed; the three other common chords of these conies will each pass through an angular point of ABC, and will all meet in a point.
CHAPTER XL
SYSTEMS OF CONICS.
204. The most general equation of a conic, viz. ow?' + 2hxy + hf+ 2gx + 2/3/ + c = 0, contains the six constants a, h, b, g, f, c. But, since we may multiply or divide the equation by any constant quantity without changing the relation between x and y which it indicates, there are really only five constants which are fixed for any particular conic, viz. the five ratios of the six constants a, h, h, g,f, c to one another.
A conic therefore can be made to satisfy five conditions and no more. For example a conic can be made to pass through five given points, or to pass through four given points and to touch a given straight line. The five con- ditions which the conic has to satisfy give rise to five equations between the constants, and five independent equations are both necessary and sufficient to determine the five ratios.
The given equations may however give more than one set of values of the ratios, and therefore more than one conic may satisfy the given conditions ; but the number of such conies will be finite if the conditions are really independent.
If there are only four (or less than four) conditions given, an infinite number of conies will satisfy them.
The five conditions which any conic can satisfy must be such that each gives rise to one relation among the constants; as, for instance, the condition of passing through a given point, or that of touching a given straight line.
232 THE CONIC THROUGH FIVE POINTS.
Some conditions give two or more relations between the constants, and any such condition must be reckoned as two or" more of the five. We proceed to give some examples.
In order that a given point may be the centre two relations must be satisfied [Art. 168].
To have a focus given is equivalent to having two tangents given [Art. 193].
To have given that a line touches a conic at a given point is equivalent to two conditions, for we have two consecutive points on the curve given.
To have the direction of an asymptote given is equiva- lent to having one point (at infinity) given.
To have the position of an asymptote given is equivalent to two conditions, for two points (at infinity) are given.
To have the axes given in position is equivalent to three conditions.
To have the eccentricity given is in general equivalent
to one condition, but since we have ^r „ = ^ i-^— 70—
1—e ah— h^
[Art. 191], if we are given that e = 0, we must have hath
a = h and h = 0.
205. Through five points, no four of which are in a straight line, one conic and only one can he drawn.
If three of the points are in a straight line, tbe conic through the five given points must be a pair of straight lines; for no straight line can meet an ^lipse, parabola, or hyperbola in three points. And the only pair of straight lines through the five points is the line on which the three points lie and the line joining the other two points.
If however not more than two of the points are on any straight line, take the line joining two of the points for the axis of x, and the line joining two others for the axis of y.
Let the co-ordinates of the four points referred to these axes be A J, 0 ; Ag, 0 ; 0,k^\ and 0, k^ respectively.
CONICS THROUGH FOUR POINTS. 233
The pairs of straight lines (^ + ^-l)(r- + ^-l)=0
and xy = 0 are conies which pass through the four points. Hence [Art. 187] all the conies given by the equation
-»-(M,-')(M.-')=»
will pass through the four points.
This conic will go through the fifth point, whose co- ordinates are x\ y\ if X be so chosen that
There is one and only one value of X which satisfies this last equation, and therefore one and only one conic will pass through the five points.
If four points lie on a straight line, more than one conic will go through the five given points, for the straight line on which the four points lie and any straight line through the fifth is such a conic.
Ex. 1. Find the equation of the conic passing through the five points
(2, 1) (1, 0), (3, -1), (-1,0) and (3, -2). The pairs of lines (cb-^/- 1) (a3 + 4y+l) = 0, and y[2x + y-b)=Q, pass through the first four points, and therefore also the conic {x-y-l){x + Ay + l)-\y {2x + y-^) = 0. The point (3, - 2) is on the latter conic if X= - 8 ; therefore the required equation is x^ + 19a;y + 4?/^ - 45^/ -1 = 0.
Ex. 2. Find the equation of the conic which passes through the five points (0, 0), (2, 3), (0, 3), (2, 5) and (4, 5).
Am. 5*2 _ lOa;^ + ^V^ + 20^ - 12y = 0.
206. To find the general equation of a conic through four fixed points.
Take the line joining two of the points for axis of a?, and the line joining the other two for axis of y, and let the lines whose equations are ax -\-hy — 1 = 0 and ax + 6'y — 1 = 0 cut the axes in the four given points.
Then xy = 0, and {ax + by — l) {a'x + fe'i/ — 1) = 0 are two conies through the four points, and therefore all the
234 CONICS THROUGH FOUR POINTS.
conies of the system are included in the equation
\xy + {ax-\-'by-l){a'x-\-h'y-V) = 0 (i),
or aaa^ + (ha + ah' ^X) xy -\- hh'y^
-(a + a')x-(h + ¥)y-\-l = 0 (ii).
207. The equation (ii), Art. 206, will represent a parabola, if the terms of the second degree are a perfect square; that is, if
^aahh' = {ha + ah' + \f.
This equation has two roots, therefore two parabolas will pass through four given points. These parabolas are real if the roots of the equation are real, which is the case when aa'hh' is positive. It is easy to shew that when aa!hh' is negative the quadrilateral is re-en- trant; in that case the parabolas are imaginary, as is geo- metrically obvious.
When the terms of the second degree in (ii). Art. 206, form a perfect square, the square must be {Jaax ± Jhb'yY. Hence [Art. 172], the axes of the two parabolas are parallel to the lines whose equations are Jaax ± sjhb'y = 0, or as one equation aax^ — hh'y^= 0.
These two straight lines are parallel to conjugate di- ameters of any conic through the four points [Art. 183].
Hence all conies through four given points have a pair of conjugate diameters parallel to the axes of the two pa- rabolas through those points.
208. To find the locus of the centres of the conies which pass through four fi^ed points.
As in Art. 206, the equation of any conic of the system is
X^y -h{ax + hy~l) {ax + h'y - 1) = 0. The co-ordinates of the centre of the conic are given by the equations
\y+a {a'x + h'y - 1) -{- a {ax -{■hy - 1) = 0, and Xx + h {ax + h'y — l)-\-h' {ax + 6^ — 1) = 0.
CONICS THROUGH FOUR POINTS. 235
Multiply these by w and y respectively and subtract ; then we have, for all values of X,
{ax — hy) {ax -\-h'y — l)-\- {ax — h'y) {ax -{- by — 1) = 0, or 2aax^ - 266y -{a + a')x + (h + h') y=0.
The locus of the centre is therefore a conic whose asymptotes are parallel to the lines aaV — 66^ = 0, i.e. parallel to the axes of the two parabolas through the four points. [The two parabolas are conies of the system, and their centres are therefore the points at infinity on the centre-locus.]
209. The centre-locus in Art. 208 goes through the origin, that is through the point of intersection of the line joining two of the points and of the line joining the other two ; and by symmetry it must go through the intersection of the other pairs of lines through the four points. [This could have been seen at once, for the pairs of lines are conies of the system and their centres are their points of intersection, and therefore these points of intersection are points on the centre-locus.]
The centre-locus cuts the axis of x where x=^0 and
where ic = J f - 4- -7 j . Therefore the locus passes through
the point midway between (-, 0) and [-, 0), that is
through the middle point of the line joining two of the fixed points, and therefore similarly through the middle point of the line joining any other two of the four points.
If then A, B,G, D he any four points, the three points of intersection of AB and CD, oi AC and BD, and oi AD and BC, together with the six middle points of AB, BC, CA, AD, BD and CD all lie on a conic, and this conic is the locus of the centres of the conies which pass through the four points A, B, C, D.
210. If aa' and hU have the same sign, we see from Art. 208 that the centre-locus is an hyperbola, and that if aa and hh' have different signs the centre-locus is an ellipse.
286 CONICS THROUGH FOUR POINTS,
If ad = hJ), that is if the four points are on a circle, the centre-locus is a rectangular hyperbola. If ad — — 66', and the axes are at right angles, all the conies of the system are rectangular hyperbolas, and the centre-locus is a circle. In this case the lines joining any two of the points is perpendicular to the line joining the other two, so that D is the ortho-centre of the triangle ABC,
Hence a circle will pass through the feet of perpendi- culars of a triangle ABG and through the middle points of AB, BC, CA, AD, BD, CD where D is the ortho-centre of the triangle ABC, and this circle is the locus of the centres of all the conies (which are all rectangular hyper- bolas) through A, B, G, J), This circle is called the nine- point circle,
211. The asymptotes of any conic through the four points defined as in Art. 206, are parallel to the lines
\xy -\- (ax + hy) (dx + h'y) = 0, or aaV + (>^ + «6' + db) xy + hh'y^ — 0.
And [Art. 183] these lines are parallel to conjugate dia- meters of the centre-locus. Hence the asymptotes of any conic through the four points are parallel to conjugate dia- meters of the centre-locus; as a particular case, the asymp- totes of the rectangular hyperbola which passes through the four points are parallel to the axes of the centre-locus.
Ex. 1. The polar of a fixed point with respect to a system of conies through four given points will pass through a fixed point. Take the fixed point for origin, and let
5 = aa;2 + 2^y + %2 + 2(7.r + 2/?/ + c = 0, and S' = a!x^ + 2 }\!xy + 6 V^ + '^q'^ + Vv + C = 0,
be two of the conies; then any conic of the system is given by /Sf- \/S" = 0. The polar of the origin is
gx +fy + c - X {g'x +fy + c') = 0, and this, for all values of X, passes through the intersection of gx +fy + c = 0, and g'x +f'y + c'=0. Ex. 2. The locus of the poles of a given straight line with respect to the conies which pass through four given points is a conic.
Take the fixed straight line for the axis of .r, and let the equation
CONICS THROUGH FOUR POINTS. 237
of any conic of the system be as in Ex. 1. The polar of (x', y') is X {ax' + hy' +g) + y{ hx' + h/ +/) + gx' +fy' + c
-\\x (aV + h'y' + g')+y {h'x' + b'y' +/') + g'x' +fij' + c' } = 0. If this is the same Hne as y = 0, the coefficient of x and the constant term must be zero. Equate these to zero and eliminate \.
Ex. 3. Shew that the locus of the pole of a given straight line with respect to any conic which passes through the angular points of a given square is a rectangular hyperbola.
[Take for axes the lines through the centre of the square parallel to the sides ; then the conies are given hj x^-a^-\ {y^ - a?) = 0.]
Ex. 4. The nine-point circles of the four triangles determined by four given points meet in a point.
212. If a = 0 and /8 = 0 are the equations of one pair of straight lines through four given points, and 7 = 0, 8 = 0 the equations of another pair, any conic through the four points has an equation of the form
a^ = %3. Now, if a = 0 be the equation of a straight line and the co-ordinates of any point be substituted in a, the result is proportional to the perpendicular distance of the point from the line. Hence the geometrical meaning of the above equation is
PxP2 '^ PJ>,> where p^, p^) P^^ P4 are the perpendiculars on the four lines a = 0, y^ = 0, 7 = 0, 8 = 0 respectively, the perpendiculars being drawn from any point on the conic.
213. If P, Q, B, Sf he four points on a conic, and QP, RS meet in A, Q8, PR in B, and PS, QR in C; then of the three points A, B, G each is the pole with respect to the conic of the line joining the other two.
Take A for origin, and the two lines ASR, APQ for axes of X and y respectively.
Let the equations of PB and QR be •
ax ^hy -1=0 (i),
a'x + h'y — 1=0 (ii).
238 SELF-POLAR TRIANGLE.
Then the equations of PR and QS will be
a'x + hy —1 = 0 (iii),
and ax +h'y —1 = 0 (iv).
The equation of any conic through the intersection of the conies xy = 0 and (ax -{-by — 1) (ax + b'y — 1) = 0, will be
Xxy + (ax + by — 1) (ax -f Ny — 1) = 0. The polar of the origin of this conic is [Art. 179] (a + a)x-\-(b + b')y-2 = 0. Writing this in the forms
ax -{-by — 1 + ax + 6'y — 1 = 0, and a'x + by — 1 i-ax +b'y — 1 = 0,
we see that the polar of the origin goes through the point of intersection of the lines (i) and (ii), and also through the point of intersection of the lines (iii) and (iv). The polar of A with respect to the conic is therefore the line BG.
It can be shewn in a similar manner that CA is the polar of B, and A B the polar of C.
A triangle which is such that each of its angular points is the pole, with respect to a conic, of the opposite side, is called a self-conjugate, or self-polar triangle.
214. If a come touch the sides of a quadrilateral and ABC be the triangle formed by the diagonals of the quadri- lateral ; then will ABC be a self -'polar triangle with respect to the conic.
Let P, Q, R, S be the points of contact.
Then, in the figure, L is the pole of F Q, and iV is the pole of SR; therefore LN is the polar of the point of
SELF-POLAR TRIANGLE.
239
intersection of PQ and SR. Similarly KM is the polar of the point of intersection of SP and B Q.
Hence A, the point of in- tersection of LN and KM, is the pole of the line joining the point of intersection of PQ, SR and the point of intersec- tipn of SP, RQ.
But [Art. 213] the point of intersection of PR and SQ h the pole of this last line.
Hence A is the point of intersection of PR and SQ.
So also B is the point of intersection of SP and RQ, and G is the point of inter- section of PQ and SR.
Hence from Art. 213 the triangle ABC is self-polar.
215. To find the general equation of a conic which touches the axes of co-ordinates.
If the equation of the line joining the points of contact be ax + by—l = 0, the equation of a conic having double contact with the conic soy = 0, where it is met by the line acc + bt/-l-= 0, is [Art. 187]
{aa; + hy- If - 2\xy = 0.
216. To find the general equation of a conic which touches four fixed straight lines.
Take two of the lines for axes, and let the equations of the other two be Ix -\- my — 1 = 0, and I'x -f my — 1 = 0. The equation of any conic touching the axes is
{ax + by-l)^-2\xy = 0 (i).
The lines joining the origin to the points where
240 CONICS TOUQHTNa FOUR LINES.
Ix + my = 1 cuts (i) are given by the equation
(ax + hy — Ix — TnyY = 2\xy (ii).
The line will touch the conic if the lines (ii) are coincident, the condition for which is
(a _ ly (h - mf = {{a -l){h- m) - Xf ;
whence \ = 2 (a — Q (& — m).
Hence the general equation of a conic touching the four straight lines
/j? = 0, 2/ = 0, lx-{- my — 1=0, and Vx + m'y —1=0, is {ax -i-hy— lY = 2Xxy ;
the parameters a, 6, X being connected by the two equations
X = 2 (a- Z) (6 - w) = 2 (a- 0 (h-m).
217. To find the locus of the centres of conies which touch four given straight lines.
If two of the lines be taken for axes, and the equations of the other two lines be
Ix + my — 1=0, and Vx + m'y — 1 = 0, the equation of the conic will be
{ax â– \-hy-Vf- 2\xy = 0, with the conditions
\=2(a-0(6-m) (i),
\=2{a-l'){h-m') (ii).
The centre of the conic is given by the equations a {ax + by — l)~\y= 0, and h {ax + by —1) —\x = 0;
,'. ax = by, and a{2ax — l) = Xy (iii).
To obtain the required locus we must eliminate a, b and \ from the equations (i), (ii), and (iii). From (i) and (iii), we have a {2ax -l) = 2y{a~ I) {b - m) = 2 (a - Z) {by - my) ; therefore, since ax = by,
a {2lx + 2my — 1) = 2lmy. Similarly, from (ii) and (iii) we have
a {2Vx + 2m y - 1) = 21! m'y.
CONICS TOUCHING FOUR LINES. 241
Eliminating a, we obtain the equation of the locus of centres, viz.
2Z;r + '^my -1 _ n'x + 2m y - 1 Im I'm
The required locus is therefore the straight line whose equation is
\m m / '^ \l 0 / Im Cm This straight line can easily be shewn to pass through the middle points of the diagonals of the quadrilateral, as it clearly should do, for any one of the diagonals is the limiting form of a very thin ellipse which touches the four lines, and the centre of this ellipse is ultimately the middle point of the diagonal. Hence the middle points of the three diagonals of a quadrilateral are points on the centre- locus of the conies touching the sides of the quadrilateral.
218. All conies 'touching the axes at the two points where they are cut by the line ax-\-hi/ — 1 = 0 are given by the equation
{ax -^hy- If = 2\xy. The conic will be a parabola if \ be such that the terms of the second degree form a perfect square : the condition for this is
a'b' = {ab-\y; .-. \=0, or \=2ah. The value \=0 gives a pair of coincident straight lines, viz. (ax -hby — Vf = 0.
Hence, for the parabola, X = 2ah, and the equation of the curve is
{ax ■\-hy — lf= 4>abxy. The above equation can be reduced to the form Jax + Jby = l.
219. To find the equation of the tangent at any point of the parabola J ax + J by = 1.
We may rationalize the equation of the curve and then S. G. s. 16
242 PARABOLA TOUCHING AXES.
make use of the formula obtained in Art. 177. The result may however be obtained in a simpler form as follows.
The equation of the line joining two points {x\ y) and {x", y") on the curve is
x" -x' y" -y ^^'
with the conditions
J^'+Jby' = l=J'^-Yjhf. (ii).
From (ii) we have
Va {^|x - s/x") ^-sjh Wy' - s/y") (iii).
Multiply the corresponding sides of the equations (i) and (iii), and we have
{x-x) = -^ ,/^ (y-y).
sjx + V^" V2/'+ ^ly
The equation of the tangent at {x ^ y) is therefore
or, since Jax' + ^hy = 1,
To find the equation of the polar of any point with respect to the conic, we must use the rationalized form of the equation of the parabola.
Ex. 1. To find the condition tJmt the line Ix + my -1 = 0 jhcuj touch the parabola J ax + Jby -1 = 0.
The equation of the tangent at any point {x\ y') is
which is the same as the given equation, if 1= /-, , and 771= . /-, ; or if - = fjax', and — = Jby'.
Hence the required condition is
a & - T + -=l. I m
\
PARABOLA TOUCHING THE AXIS. 243
Ex. 2. To find the focus of the parabola whose equation is
Jax + Jlnj = l. The circle which touches TQ at T and which passes through P will also pass through the focus [see Art. 165 (4), two of the tangents being
coincident]. The two points P, Q are ( - > 0 j and ( 0, - ] . Therefore
the focus is on both the circles whose equations are
x^ + 2xy cos o) + ?/2 — = 0, a
and x^ + 'ixycoQo}^y--j = 0.
Hence the focus is given by
Ex. 3. To find the directrix of the parabola isj ax + fj by = 1. The directrix is the locus of the intersection of tangents at right angles; now the line lx + my = l will be perpendicular to y = 0 if
m-l cos w = 0, and the line will touch if - + — = 1. Therefore the inter-
l m
cept on the axis of x made by a tangent perpendicular to that axis is
given by y(a + )=1.
I \ cos wy
Hence the point ( ^j , 0 ) is on the directrix.
\6 + acosw /
Similarly the point ( 0, • , — ) is on the directrix. ^ V « + 6cosw/
Hence the required equation is
x{b + a cos u))-\-y {a-\-h cos w) = cos w.
220. Since the foci of a conic are on its axes, if two conies are confocal they must have the same axes.
The equation
will, for different values of X, represent different conies of a confocal system. For the distance of a focus from the centre is
^[{a"- + X) - (6' + X)] or sl{d' - h\
16—2 •
244
CONFOCAL CONICS.
221. The equation of a system of confocal conies is
= 1.
If X is positive the curve is an ellipse.
The principal axes of the curve will increase as X increases, and their ratio will tend more and more to equality as X is increased more and more; so that a circle of infinite radius is a limiting form of one of the confocals.
If X be negative, the principal axes will decrease as
6^ 4- X X increases, and the ratio -„-— ^ will also decrease as X
a^ + X
increases, so that the ellipse becomes flatter and flatter, until X is equal to — h^, when the minor axis vanishes, and the major axis is equal to the distance between the foci. Hence the line-ellipse joining the foci is a limiting form of one of the confocals.
li¥-\-X is negative, the curve is an hyperbola.
CONFOCAL CONICS. 245
If 6^ 4- X is a sifYiall negative quantity the transverse axis of the hyperbola is very nearly equal to the distance between the foci ; and the complement of the line joining the foci is a limiting form of the hyperbola.
The angle between the asymptotes of the hyperbola will become greater and greater as —X- becomes greater and greater and in the limit both branches of the curve coincide with the axis of y.
If X is negative and numerically greater than a^, the curve is imaginary.
222. Two conies of a confocal system pass through any given point. One of these conies is an ellipse and the other an hyperbola.
Let the equation of the original conic be
The equation of any confocal conic is
a^ + X + fti^+X ^• This will pass through the given point (&', y), if
+ w^r^=l.
a'^+X 6' + X
In the above put Z>^+ X = X' ;
then x"^ + y'^ (V + a V) - V (X' + aV) = 0,
or V' - X' {al^ + y"" - aV) - aVy' = 0.
The roots of this quadratic in X' are both real, and are of different signs. Therefore there are two conies, and b^ + \ is positive for one, and negative for the other, so that one conic is an ellipse and the other an hyperbola.
223. One conic of a confocal system and only one will touch a given straight line.
Let the equation of the given straight line be Ix -h my — 1 = 0.
246 CONFOCAL CONICS.
The line will touch the conic whose equation is
.2 , ^ ~ 12 . ^ -^>
a' + X 6' + X
if {a^ + X) Z« + (6' + X) m' = 1 [Art. 115],
which gives one, and only one, value of X. Hence one confocal will touch the given straight line.
224. Two confocal conies cut one another at Hght angles at all their cornmon points.
Let the equations of the conies be
^, + ^=l,and^,--^ + ^,^^ = l,
and let {cc\ yf) be a common point ; then the co-ordinates x\ y will satisfy both the above equations.
Hence, by subtraction, we have
a^(a^ + X) ' h\h^+\)
Now the equations of the tangents to the conies at
{x, y) are
—2 + T2 = 1. and ., — + ,.f f ^^ = 1
respectively.
The condition (i) shews that the tangents are at right angles to one another.
225. The difference of the squai^es of the perpendiculars drawn from the ce^itre on any two parallel tangents to two given confocal conies is constant.
Let the equations of the conies be
x' y^ , ^ x"- v' .
-2 +f^ = 1, and -2-r^ + ifv-:"^ ^' a 0 a -hX 6 +X
Let the two straight lines
X cos 0L-\- y sin a — p = 0, x cos a -f i/ sin a—p'= 0,
CONFOCAL CONICS. 247
touch the conies respectively ; then [Art. 115, Cor.] we
have p^ = a^ cos^ a + J^ sin^ a,
and ^/' = (a' + X) cos' a + {!)' + X) sin'^a ;
226. If a tangent to one of two confocal conies he perjpendicidar to a tangent to the othe7% the locus of their point of intersection is a circle.
Let the equations of the confocal conies be
2 2 2 2
^ + 'f? =1, and 2-— + -^ - 1
a' ' b' ' a'-\-\ ' h'' + \
The lines whose equations are
x cos oi+ y sin a = */{d^ cos' ol + ¥ sin' a) (i),
a; sin a - 2/ cos a = ^[{a^ -f X) sin^ a + (Z>' + X) cos' a} . . . (ii), touch the conies respectively, and are at right angles to one another.
Square both sides of the equations (i) and (ii) and add, then we have for the equation of the required locus x' + y' = a'+Â¥ + \.
If we suppose the minor axis of the second ellipse to become indefinitely small, all tangents to it will pass indefinitely near to a focus; so that Art. 125 {tj) is a particular case of the above.
Ex. 1. Any two parabolas which have a common focus and their axes in opposite directions intersect at right angles.
Ex. 2. Two parabolas have a common focus and their axes in the same straight line ; shew that, if TP, TQ be tangents one to each of the parabolas, and TP, TQ be at right angles to one another, the locus of T is a straight line.
Ex. 3. TQ, TP are tangents one to each of two confocal conies whose centre is C ; shew that if the tangents are at right angles to one another CT will bisect PQ.
Let the tangents be
XX' 7/?/' ^ , X3f' VV" -,
248 CONFOCAL CONICS.
the equation of CT will be
This will pass through the middle point of PQ, if
that is, if
^'^'(a'^-i)+2/y'(p-p)=0;
or, since the conies are confocal, if
That is, if the tangents are at right angles.
Ex. 4. TP, TQ are tangents one to each of two parabolas which have a common focus and their axes in the same straight line ; shew that, if a line through T parallel to the axis bisect PQ, the tangents will be at right angles.
Ex. 5. If points on two confocal ellipses which have the same eccen- tric angles are called corresponding points ; shew that, if P,Q be any two points on an ellipse, and p, q be the corresponding points on a confocal ellipse, then Pq = Qp.
227. The locus of the pole of a given straight line with respect to a series of confocal conies is a straight line.
Let the equation of the confocals be
'cF+\'^W+\^^ ^'^'
and let the equation of the given straight line be
Ix+my = 1 (ii).
The equation of the polar of the point {x\ y) with
respect to (i) is -,-^ + ^.^ = 1 (m).
If (ii) and (iii) represent the same straight line, we
must have —5 — - = I, ,/ ^ = m ;
a + X h +\
:. ^-a' = l--b' = \. I m
CONFOCAL CONICS. 249
Hence the locus of the poles is the straight line whose equation is
^ y 2 72
T — — = a — 0 . I m
This straight line is perpendicular to the line (ii). One confocal of the system will touch the line (ii), and the point of contact will be the pole of the line with respect to that confocal.
Hence the locus of the poles is a straight line perpen- dicular to the given straight line and through the point where it touches a confocal.
228. From any point T the two tangents TP, TP' are drawn to one conic, and the two tangents TQ^ TQ' to a con- focal conic; shew that the straight lines QP, Q'P will make equal angles with the tangent at P.
Let TP and the normal at P cut QQ' in Ky L respectively.
Then [Art. 227] the pole of TP, with respect to the conic on which Q, Q' lie, is on the line PL. Also, since T is the pole of QQ' with respect to that conic, the pole of TP is on QQ' [Art. 180]. Therefore the pole of TPK is at L, the point of intersection of QQ' and PL.
Therefore [Art. 181] the range K, Q, L, Q\ and the pencil PK, PQ, PL, PQ\ are harmonic.
250 CONFOCAL CONICS.
Hence, since the angle KPL is a right angle, P Q and PQ make equal angles with PL or PK [Art. 56].
Cor. 1. Let the conic on which Q, Q' lie degenerate into the line-ellipse joining the foci, th«n the proposition becomes — The lines joining the foci of a conic to any point P on the curve make equal angles with the tangent at P.
Cor. 2. Let the conic on which P, P' lie degenerate into the line-ellipse, and we have — Two tangents to a conic subtend equal angles at a focus.
Cor. 3. Let the conic on which P, P' lie pass through T, and we have — The two tangents drawn to a come from any point T make equal angles tuith the tangent at T to either of the confocal conies which pass through T.
229. If QQ' he any chord of a given conic which touches a fixed confocal conic, then will QQ' vary as the square of the parallel diameter. Also, if GE he drawn through the centre parallel to the tangent at Q and Tmeting QQ' in E, then will QE he of constant length.
Let The the pole of QQ\ and let CT cut QQ in F, and the curve in P. Also let CD be the semi-diameter paral- lel to Qq.
Let p, p be the lengths of the perpendiculars from the centre on QQ\ and on the parallel tangent to the ellipse QPQ' ; then [Art. 225] we have
CONFOCAL CONICS. 251
,2 7
uence ^, - i ^, - i ^^, - ^^, therefore, since jp . CD = a6, we have
P
a'i"
,.Qr=^oi)^ (i).
Also QE^ GT _ CV. CT GP^ _GD^
QV" Vf~ GV. GT- GV" GP'- GV'~ QV' therefore from (i) we have
m? (7I>^ ah ,...
«^=-^=v^ (")•
Ex. TP, TQ are tangents one to each of two fixed confocal conies ; shew that, if the tangents are at right angles to one another, the line PQ will always touch a third confocal conic.
If G be the common centre, then since the tangents are at right angles to one another the line CT bisects PQ [Ex. (3) Art. 226]. Therefore GT and QP make equal angles with the tangent at Q. If therefore GE be parallel to the tangent at Q, and meet QP in E, we have QJE—CT.
But CT is constant [Art. 226]. Hence QE is constant, and therefore QEP touches a fixed confocal.
230. When two of the points of intersection of any two curves are coincident, that is when the two curves touch, they are said to have contact of the first order at the point. When three points of intersection are coincident the curves are said to have contact of the second order, and so on.
A curve which has with a given curve a contact of the highest possible order is called an osculating curve.
A circle can only be made to pass through three given points ; hence the circles which osculate a curve have contact of the second order mth it.
The circle which has contact of the second order with a giveii curve at a given point is generally called the circle
252 THE CONTACT OP CONICS.
of curvature at that point, and the radius of the circle is called the radius of curvature at the point.
Two conies intersect in four points. Hence two conies cannot have contact with one another of higher order than the third. If they have contact of the second order they will have one other common point.
231. To find the general equation of a conic which has contact of the second order with a given conic at a given 'point
Let ^ = 0 be the equation of the given conic, and let ^ = 0 be the equation of the tangent to yS' = 0 at the given point {x, y').
The equation of any straight line through {x\ y') is y — y—ni{x— x) = 0.
Hence the equation
8-\T{{y-y')-m{x-x')]=^^ ......{i)
is the equation of a conic passing through the points where the straight lines T = 0, and y— y —m {x — x') = 0 cut
Hence (i) intersects S=0 in three coincident points. The two constants X and m being arbitrary, the conic given by (i) can be made to satisfy two other conditions. They can for instance be so chosen that the equation (i) shall represent a circle.
If the line y — y' — m {x —x') = 0 coincides with the tangent, all four points of intersection are coincident. The conic S—\T^ = 0 therefore has contact of the third order with S = 0 ; that is to say, is the equation of an oscidating conic.
Ex. 1. Find the equation of the circle which osculates the conic ax^ + 2hxy + cy^ + 2dx = 0 at the origin. All the conies included in the equation
az^ + 2hxy + cy"^ + 2dx - \x [y — mx) =0 have contact of the second order.
The conditions for a circle are 26-X=:0 and a + \m--c. Therefore the circle required is cx^-\-cy^ + 2dx=Q.
THE CONTACT OF CONICS. 253
Ex. 2. Find the equation of the parabola which has contact of the third order with the conic ax^ + 2hxy + cy^ + 2dx=0 at the origin.
The conic ax^ + 2bxy + cy^ + 2dx-\x^ = 0 cuts the given conic in four coincident points.
The curve is a parabola if (a - X) c = b\
The equation of the required parabola is therefore b^X'^ + 2bcxy + c'^f + 2dcx = 0.
232. Since the line joining any two of the points of intersection of a circle and a conic, and the line joining the other two points of intersection, make equal angles with the axis of the conic, we see that, if the circle of curvature at a point P of a conic cut the conic again in 0, the tangent at P and the chord P 0 make equal angles with an axis of the conic.
233. If a, fi, y, S be the eccentric angles of four points on an ellipse, a circle will pass through those four points, if
a-\-^ + y + B=2n7r [Art. 184. Ex. 1].
Hence the circle of curvature at the point a will cut the ellipse again at the point 8 where
3a + S = 2^i7r (i).
From (i) we see that, through any particular point 8 three circles of curvature will pass, viz. the circles of curvature at the points J (27r — S), J (47r — 8), and J {Qtt — 8). These three points are the angular points of a maximum triangle inscribed in the ellipse [Art. 138 (1)]. Also, since 3 + i(27r-S)+|-(47r-S) + i(67r-S) = 47r, the point 8 and the three points the circles of curvature at which pass through 8 are on a circle.
234. To find the equations of the three pairs of straight lines which can he drawn through the points of intersection of two conies.
Let the equations of the conies be
S=ax^-Â¥2hxy-\-hif + 2gx-\-yy+c=0,
and 8' = ax''+2h'xy-^Vy^-\-2g'x+2fy 4- c' = 0.
254 INVARIANTS.
The equation of any conic through their points of in- tersection is of the form
S+XS' = 0 (i).
The conic S + XiS' = 0 will be a pair of straight lines,
if
0 (ii).
a + W, h + \h\ g + \g' h + \h\ h + \h\ f-\-Xf g + \g\f-\-\f\ c-^Xc'
We have therefore a cubic for the determination of \. If any root of this cubic be substituted in (i) we have the equation of one of the three pairs of straight lines.
If X be eliminated between the equations (i) and (ii) we have an equation of the sixth degree which represents the three pairs of straight lines.
Since one root of a cubic equation is always real, one value of X is in all cases real.
It can be shewn that at least one pair of straight lines is in all cases real. [See Salmon's Conies, Art. 282.]
235. The equation (ii) Art. 234 is usually written
A + X0 + X'^@'-|-XW = O.
If the axes be changed in any manner, and the equa- tions of the two conies become X = 0 and %' = 0, the equa- tion S + \S' = 0 will become S + X2' = 0 ; and if X be such that S + \S'=0 represents a pair of straight lines, so also will % + XS' = 0. Hence the values of X for which S + X^' = 0 represents straight lines must be independent of any particular axes of co-ordinates ; hence the ratios of the four quantities A, ©, 0', A' to one another must be independent of the axes of co-ordinates. For this reason they are called the Invariants of the system. The student will find interesting applications of invariants in Salmon's Conic Sections and Wolstenholme's Problems.
236. We shall conclude this Chapter by the solution of some Examples.
EXAMPLES.
255
Ex. 1. If two conies have each double contact with a third, their chords of contact with that conic, and tivo of the lines through their common points, will meet in a point and form a harmonic pencil.
Let /5=0 be the equation of the third conic, and let a = 0, /3 = 0 be the
equations of the two chords of contact. Then [Art. 187] the equations of
the conies are
^-X2a2 = 0 (i),
and S-fj?^^=0 (ii).
Now the two straight lines
\^a^-fi^^ = 0 (iii)
go through the common points of (i) and (ii). The lines (iii) also go through the point of intersection of a = 0 and /8=0 ; and [Art. 56] the four lines a = 0, \a-fip = 0, /3 = 0, and Xa + /x/3 = 0 form a harmonic pencil.
Ex. 2. A circle of given radium cuts an ellipse in four points ; shew that the continued product of the diameters of the ellipse parallel to the common chords is constant.
Let the equation of the ellipse l>e -^ + ^ = 1, and the equation of the
Then the equation of any pair of
circle be {x~af + [y-^f-k^^(i. common chords is
(a;-a)2 + (y-^)^-fc«-x(^ + ^-l)=0.
.(i),
where X is one of the roots of the equation X
0,
0,
1-^
6^:
= 0.
.(ii)
The equation of the diameters of the ellipse parallel to the lines (i) is a^'^ + 2/^-x(^' + ^')=0 (iii).
The two semi-diameters given by (iii) clearly make equal angles with the axis, and the square of the length of one of them is equal to X.
Hence the continued product of the six semi-diameters is equal to the product of the three values of X given by (ii), which is easily seen to be a%^k^.
Ex. 3. If a conic Jiave any one of four given points for centre, and the triangle formed by the other three for a self polar triangle, its asymptotes will he parallel to the axes of the two parabolas lohich pass through the four points.
256 EXAMPLES.
Let the four points be given by the intersections of the straight lines
xy = 0 and {Ix + my-l) {I'x + m'y - 1) = 0. The line joining the centre of a conic to any one of the angular points of a self polar triangle is conjugate to the line joining the other two angular points. Hence, for all the four conies, the three pairs of lines joining the four given points are parallel to conjugate diameters. Let the equation of one of the conies be
ax"^ + 11ixy + 1)11' + 1gx + 2fy + c = Q (i).
The lines ( Ix + my -l){l'x + m'y - 1 ) = 0
are parallel to conjugate diameters ; therefore also the lines
IVx^ + {Im' + I'm) xy + mm'y^ = 0 are parallel to conjugate diameters. Hence [Art. 183], we have am.vi' + hlV = h {Im' + I'm). The lines xy = 0 are parallel to conjugate diameters; therefore h=0, and we have
avim'+hll'=0 (ii).
The asymptotes of (i) are parallel to the straight lines ax^ + by^ = 0, or, from (ii), the asymptotes are parallel to the lines
ll'x^ - mm'y^ = 0, which proves the theorem [Art. 207].
Ex. 4. The circumscribing circle of any triangle self polar with respect to a conic cuts the director-circle orthogonally.
Let the equation of the conic be ax^ + hy^ = l ; and let (a/, y'), {x", y") and («'", y'") be the angular points of the triangle.
Since each of the points is on the polar of another, we have
ax"x"'+l)y"y"'-l=^0 (i),
ax"'x' + lyy'Y -1=0 (ii),
and ax'x" + l)y'y" -1 = 0 (iii).
The equation of the circle circumscribing the triangle is
x^ + y^, X, y, 1 =0 (iv).
a/2 + 2^'2, x', y', 1 x"^ + f\ x", y", 1 x"'^ + f'\ x'", y'", 1 Now, if the equation of a circle be
Ax'^-\-Ay^ + 2Gx + 2Fy + G = 0, the square of the tangent to it from the origin is equal to the ratio of C to A.
Hence the square of the tangent to the circle (iv) is equal to the ratio of
EXAMPLES.
I a;'2 + y'2, x\ y' to- x, y\ 1 «"2 + y"2, x", y" x", y'\ 1
U"^^r\ «'". y" «'"', 2/'". 1
The first determinant is equal to ^ (a/y " - yV") + a/'a (X'Y - f'x') + x'"2 (xy ' - y V)
+ y'a(x'y"-y'V") + y'M^"y-y"V)+y'"Ma:y'-/x") (a)
Now from the equations (i), (ii), (iii) we have ox' hy' - 1
and
257
|
y"'-y" ax" |
-x"-x'" _ %" ~x"'-x' hy"' = x'-x" |
-.'Y |
-1 |
|
y'-y'" ax'" |
~ a;'y"' |
-2/V'" 1 |
|
|
y"-y' |
^Y'- |
-y'V |
By means of these equations, (a) becomes
(/
2/")+-(!/'-y"')+-(y'
2/0
+ 1 (x"-x"') + |- (x"'-a:') +2^ {x'-x").
Gn)
a;', 2/', 1 ^", 2/", 1 x'", y'", 1
Hence the tangent to the circumscribing circle from the centre of the conic is equal to A/(- + r)» t^^^* ^^ equal to the radius of the director- circle, which proves the proposition.
Examples on Chapter XI.
1. Two sfci^ight lines of given length are moved along two given straight lines in such a manner that a circle will pass through their four extremities; shew that the locus of the centre of this circle is a rectangular hyperbola.
2. OPP', OQQ' are two chords of a conic, and any line through 0 cuts the conic in R, R' and the lines PQ^ P'Q' in aS, aS'; shew that
1111
OR^ OK
OS^ OS'
S. 0. s.
17
258 EXAMPLES ON CHAPTER XI.
3. A system of conies pass through the same four points and the tangent at a given point 0 of one of the conies cuts any
other of the conies in P, P' ; shew that ^^p + 7777/ is constant.
4. A circle and a rectangular hyperbola intersect in four points, and one of their common chords is a diameter of the hy- perbola; shew that the other chord is a diameter of the circle.
5. Of all conies which pass through four given points that which has the least eccentricity has its equi-conjugate diameters parallel to the axes of the two parabolas through the points.
6. Of all conies which touch two given straight lines at given points the one of least eccentricity will be that in which one of the equi-conjugate diameters passes through the intersec- tion of the given lines.
7. The locus of the middle point of the intercept of a variable tangent to a conic on two fixed tangents OA, OB is a conic which reduces to a straight line if the original conic is a parabola.
8. Two tangents OA, OB are drawn to a conic and are cut in P and ^ by a variable tangent; prove that the locus of the centre of the circle described about the triangle OPQ is an hyperbola.
9. A conic is drawn touching the co-ordinate axes OX, OY Sit A, B and passing through the point J) where OA DB is a parallelogram ; shew that if the area of the triangle OAB be constant, the locus of the centre of the conic will be an hyperbola.
10. Tangents are drawn from a fixed point to a system of conies touching two given straight lines at given points. Prove that the locus of the point of contact is a conic.
11. Shew that the locus of the pole of a given straiglit line with respect to a series of conies inscribed in the same quadrilateral is a straight line.
12. An ellipse is described touching the asymptotes of an hyperbola and meeting the hyperbola in four points; shew that two of the common chords are parallel to the line joining the points of contact of the ellipse with the asymptotes, and are equidistant from that line.
EXAMPLES ON CHAPTER XI. 259
13. In a system of conies which have a given centre and their axes in a given direction, the sum of the axes is given ; shew that the locus of the pole of a given straight line is a parabola touching the axes.
14. A parabola is drawn so as to touch three given straight lines ; shew that the chords joining the points of contact pass each through a fixed point.
15. Shew that, if a parabola touch two given straight lines, and the line joining the points of contact pass through a fixed point, the locus of the focus will be a circle.
1 6. If the axis of the parabola ^| ax + Nhy - 1 pass through a fixed point, the locus of the focus will be a rectangular hyperbola.
17. From a fixed point 0, a })air of secants are drawn meeting a given conic in four points lying on a circle; shew that the locus of the centre of this circle is the perpendicular from 0 on the polar of 0.
18. TP^ TQ are tangents to a conic, and R any other point on the curve; RQ, RP meet any straight line through T in the points K^ L respectively; shew that QL and PK intersect on the curve.
19. Any point P on a fixed straight line is joined to two fixed points JL, ^ of a conic, and the lines PA, PB meet the conic again in Q^ R; shew that the locus of the point of inter- section of BQ and AR isa conic.
20. The confocal hyperbola through the point on the
x' 7/ ellipse -2 + j-2 = ^ whose eccentric angle is a has for equation
a' b'
y'
21. Find the locus of the points of contact of tangents to a series of confocal conies from a given point in the major axis.
22. If X, /A be the parameters of the eonfocals which pass through two points /J ^ on a given ellipse; shew (i) that if P, Q be extremities of conjugate diameters then A. + /x is con- stant, and (ii) that if the tangents at P and Q be at right angles
then - + - is constant.
17—2
260 EXAMPLES ON CHAPTER XL
23. Shew that the ends of the equal conjugate diameters of a series of confocal ellipses are on a confocal rectangular hyperbola.
24. Find the angle between the two tangents to an ellipse from any point in terms of the parameters of the confocals through that point ; and shew that the equation of the two tangents referred to the normals to the confocals as axes will be
25. The stmight lines OPr, OQQ' cut an ellipse in P, P' and Qy Q' respectively and touch a confocal ellipse; prove that OP . OP' . QQ' = OQ . OQ' . PF.
26. The locus of the points of contact of the tangents drawn from a given point to a system of confocals is a cubic curve, which passes through the given point and through the foci.
27. Shew that the locus of the points of contact of parallel tangents to a system of confocals is a rectangular hyperbola; and the locus of the vertices of these hyperbolas for all possible directions of the tangent is the curve whose equation is
28. If a triangle be inscribed in an ellipse and envelope a confocal ellipse, the points of contact will lie on the escribed circles of the triangle.
29. If an ellipse have double contact with each of two confocals, the tangents at the points of contact will form a rectangle.
30. If from a fixed point tangents be drawn to one of a given system of confocal conies, and the normals at the points of contact meet in Q^ shew that the locus of ^ is a straight line.
31. A triangle circumscribes an ellipse and two of its angular points lie on a confocal ellipse ; prove that the third angular point lies on another confocal ellipse.
32. An ellipse and hyperbola are confocal, and the asymp- totes of the hyperbola lie along the equi conjugate diameters of the ellipses; prove that the hyperbola will cut at right angles all conies which pass through the ends of the axes of the ellipse.
EXAMPLES ON CHAPTER XI. 261
33. Four normals are drawn to an ellipse from a point F ; prove that their product is
where A,, \^ are the parameters of the confocals to the given ellipse which pass through F and a,bthe semi-axes of the given ellipse.
34. Shew tliat the feet of the perpendiculars of a triangle are a conjugate triad with respect to any equilateral hyperbola which circumscribes the triangle.
35. 2^F, TQ are the tangents from a point T to b. conic, and the bisector of the angle FTQ meets FQ in 0 ; shew that, if FOU' be any other chord through 0, the angle BTF' will be bisected by OT.
36. If two parabolas are drawn each passing through three points on a circle and one of them meeting the circle again in J), the other meeting it again in JE; prove that the angle between their axes is one-fourth of the angle subtended by 3U at the centre of the circle.
37. If ABC be a maximum triaugle inscribed in an ellipse and the circle round ABC cut the ellipse again in D; shew that the locus of the point of intersection of the axes of the two parabolas which pass through A^ B^ (7, i> is a conic similar to the original conic.
38. If any point on a circle of radius a be given by the co-ordinates a cos B, a sin 6] shew that the equations of the axes of the two parabolas through the four points a, ^, y, S are
. ^ a fcos (S-a) + cos (S - B) + cos (S- y)) ^ 4 (+cos(aS'-8) r
. ci a <^ fsin (*S' - a) + sin (S - B) + sin (aS' - y)\ ,
4 (+sin (o-d) j
where 4:S=a+p + y + 8.
If the axes of the two parabolas intersect in F, shew that the five points so obtained by selecting four out of five points on
the circle in all possible ways, lie on a circle of radius j .
262 EXAMPLES ON CHAPTER XI.
39. li A, B, C, D he the sides of a quadrilateral inscribed in a conic, the ratio of the product of the perpendiculars from any point F of the circle on the sides A and C to the product of the perpendiculars on the sides B and D will be constant. Shew also, that if A, B, C, D, E, F...he the sides of a polygon inscribed in the conic, the number of sides being even, the continued product of the perpendiculars from any point on the conic on the sides A, C, -£',... will be to the continued product of the perpendiculars from the same point on the sides B, i>, i^,...in a constant ratio.
40. 0 is the centre of curvature at any point of the ellipse -g + ?2 = 1 ; Qi ^ are the feet of the other two normals
Cl 0
drawn from 0 to the ellipse ; prove that, if the tangents at Q and R meet in T, the equation of the locus of 1^ is —^+ -^ = 1.
41. Shew that a circle cannot cut a parabola in four real points if the abscissa of its centre be less than the semi-latus rectum.
A circle is described cutting a parabola in four points, and through the vertex of the parabola lines are drawn parallel to the six lines joining the pairs of points of intersection; shew that the sum of the abscissae of the points where these lines cut the parabola is constant if the abscissa of the centre of the circle is constant.
42. Three straight lines form a self-polar triangle with respect to a rectangular hyperbola. The curve being supposed to vary while the lines remain fixed, find the locus of the centre.
43. If a circle be described concentric with an ellipse, shew that an infinite number of triangles can be inscribed in
the ellipse and circumscribed about the circle, if - = - + t » ^ cab
where c is the radius of the circle, and a, h the semi-axes of
the ellipse.
44. Find the points on an ellipse such that the osculating circle at P passes through ^, and the osculating circle at Q passes through P,
EXAMPLES ON CHAPTER XI. 263
45. Prove that the locus of the centres of rectangular hyperbolas which have contact of the third order with a given parabola is an equal parabola.
46. P, Q are two points on an. ellipse: prove that if the normal at P bisects the angle that the normal at Q subtends at P, the normal at Q will bisect the angle the normal at P subtends at Q.
47. Shew that the centre of curvature at any point P of an ellipse is the pole of the tangent at P with respect to the confocal hyperbola through P.
48. ABC is a triangle inscribed in an ellipse. A confocal ellipse touches the sides in A\ B', C. Prove that the confocal hyperbola through A meets the inner ellipse in ^'.
49. Of two rectangular hyperbolas the asymptotes of one are parallel to the axes of the other and the centre of each lies on the other. If any circle through the centre of one cut the other again in P, Q, P, then PQP will form a conjugate triad with respect to the first.
50. A circle through the centre of a rectangular hyperbola cuts the curve in the points A, B, C, D. Prove that the circle circumscribing the triangle formed by the tangents at ^, ^, C passes through the centre of the hyperbola and has its centre at the point on the hyperbola diametrically opposite to D.
CHAPTER XII.
ENVELOPES.
237. In the general equation of a straight line the
two constants are not in any way connected. If however
the two constants are connected by any relation, the
equation will no longer represent any straight line. We
have seen, for example, that if the constants I and m
in the equation Ix + my — 1 = 0 satisfy the equation
a^r+Z>W=l, where a and h are known, the line will always
x^ v^ touch the ellipse -2 'f *y2 = 1 [A^^- H^]. In every such
case, in which the two constants in the equation of a straight line are connected by a relation, the line will touch some curve. This curve is called the envelope of the moving line.
By means of the relation connecting the two constants we may eliminate one of them, and the equation of the straight line will then contain only one indeterminate constant. If different values be given to this constant we shall have a series of different straight lines all of which will touch some curve.
238. To find the envelope of a line whose equation contains an indeterminate constant of the second degree.
Write the equation of the line in the form
fi'P+2fiQ + R = 0 (i).
where //. is the constant.
ENVELOPES. 265
Through any particular point two of the lines will pass, for if the co-ordinates of that point be substituted in (i), we shall have a quadratic equation to determine fi. Now the two tangents through any point will be coincident, if the point be on the curve which is touched by the moving line.
Hence, to find the equation of the envelope, we have only to write down the condition that the roots of (i) may be equal, viz. Q^-FR = 0.
Ex. 1. To find the envelope of the line
a ^ m
The equation may be written vi^x-my-\-a=Qy and the condition for equal roots gives 2/2 = 4aa;.
Ex. 2. Find the envelope of a line which cuts off from the axes intercepts whose sum is constant.
If the equation of the line be - + v = l, we have A + A; = constant = c. h k
Therefore v H — ^ = 1, or h^ - h {x - y + c) + occ=0. Whence the equation
of the envelope is 4iCx =(x-y + c)^.
Ex. 3. Find the envelope of the line ax cos d + hy sin d = c. The equation is equivalent to
> a
ax-c + 2byt- {ax + c)t^ — 0, where t = tan-. Hence, the envelope is
{ax-c){ax + c) + hY = 0, or a^x^ + bY = c^'
Ex. 4. 2%e envelope of the polar of a given point 0 with respect to a system of confocal conies is a parabola whose directrix is CO, where C is the centre of the confocals.
If the confocals be given by the equation
-^ + J^=l a^ + \^b^ + \ '
and 0 be the point {xf, y'), the line whose envelope is required is given by
XX' yy' _
a^ + X^lr' + X' '
or by X2 - X {x'x + y'y -a^- b^) + aW - 6 Va; - ah/y = 0,
266 ENVELOPES.
The equation of the envelope is therefore
4 {a?h^ - iVx - ah/y) = {x'x + y'lj -a?~ h'^Y.
The envelope is therefore a parabola.
Two confocals pass through 0, and the polars of 0 with respect to them are the tangents at 0; hence, since these tangents are at right angles to one another, the point 0 is on the directrix of the parabola. By considering the limiting forms of the confocals as in Art. 221, we see that the axes themselves are polars of 0 ; hence G is on the directrix of the parabola; so that the directrix is the line CO.
239. To find the envelope of the line Ix + my + 1=0, where
ar + 2hlm + hm^ + 2gl + 2/m -h c = 0.
If the line pass through a particular point {x\ y') we have Ix + my' +1 = 0. Using this to make the given con- dition homogeneous in I and m, we have the equation aT^ + 2AZm + 3w'- 2 {gl +fm) {Ix + my') + c [Ix + mij'f = 0.
The two values of the ratio — give the directions of
m ^
the two lines which pass through the point {x, y').
If {x\ y) be a point on the curve which is touched by the moving line, the tangents from it must be coincident, and therefore the above equation must be a perfect square. The condition for this is
(a - 2gx' + ex' ) {h - 2fy' + cy") = {h - gy' -fx + cx'yj, which reduces to ^.- (6c -f) + 2xY {fg - ch) + y-- [ca - g^)
+ 2x' {fh - gh) + ^' {gh -fa) +ab-Â¥=:0. The required envelope is therefore the conic Ax' + 2Hxy + By'-^2 Gx-{- 2Fy + G= 0, where A, B, (7, F, G, H mean the same as in Art. 178.
The condition that Ix + viy + 1 = 0 may touch Ax^ + 2Hxy + By^ + 2Gx + 2Fy + C = 0 is aZ^ + 2hlm + bm^ + 2gl + 2fm + c = 0. Hence by comparing with the condition found in Art. 178, we see that a, h, c, &c. must be proportional to the minors of A, B, C, &c. in tlie determinant
TANGENTIAL EQUATIONS. 267
A H G H B F
G F C
This is easily verified, for the minor of ^ is JSC- J^^^ o,. {ca - rjf2) [ab - 7i2) - (gh - of f, that is a A ; and so for the others.
Ex. 1. To find the envelope of the line lx + my + l = Q where
The directions of the lines through (x, y) are given by
aP + lm^ + c {Ix + myf = 0. These lines will coincide, if
{a + cx^){b + cy^) = c^xY' Hence the equation of the envelope is
- + v + -=o.
a 0 c Ex. 2. To find the envelope of the line Ix + my +1 = 0 with the condition
l + l + h = 0. I m
The directions of the two lines through [x, y) are given by
him - {fm + gl) {Ix + my) = 0.
They will therefore coincide if
^ffJ^j={fx+gy-h)K
This is equivalent to
240. If the equation of a straight line be Ix + my + 1 = 0, then the position of the line is determined if I, m are known, and by changing the values of I and m the equation may be made to represent any straight line whatever. The quantities I and m which thus define the position of a line are called the co-ordinates of the line.
If the co-ordinates of a straight line are connected by any relation the line will envelope a curve, and the equation which expresses the relation is called the tan- gential equation of the curve.
If the tangential equation of the curve is of the ?ith degree, then n tangents can be drawn to the curve from any point.
268 DIRECTOR-CIRCLE OF ENVELOPE.
Def, A curve is said to be of the nth class when n tangents can be drawn to it from a point.
We have seen [Art. 239] that every tangential equation of the second degree represents a conic; also [Art. 178] that the tangential equation of any conic is of the second degree.
If the equation of a straight line be Ix 4- 'my + w = 0, we may call Z, m, n the co-ordinates of the line ; and if the co-ordinates of the line satisfy any homogeneous equation, the line will envelope a curve, of which that equation is called the tangential equation.
241. To find the director-cirxle of a conic whose tangential equation is given.
Let the tangential equation of the conic be
aP + 2hlm + hm^ + 2gl + 2fm + c = 0.
As in Art. 239, the equation
al^ + 2hlm -f bni"- 2 {gl -{-fm) {Ix + mi/)+c (Ix + myf = 0
gives the directions of the two tangents which pass
through the particular point (iP, y). These tangents will
be at right angles to one another if -* -^ 4- 1 = 0, that is, ® ° m^ m^
if the sum of the coefficients of P and m^ is zero.
If therefore {x, y) be a point on the director-circle of the conic, we shall have
a-2gx + ex" +h -2fy + cy^ = Q (i).
The centre of the conic, which coincides with the centre
of the director-circle, is the point ( > ) •
If c = 0, the equation (i) is the equation of a straight line. The curve is in this case a parabola, and the equation of its directrix is
2gx + 2fy-a-h=0 (ii).
In the above we have supposed the axes to be rect- angular ; if, however, the axes of co-ordinates are inclined
FOCI OF ENVELOPE. 269
to one another at an angle o), the condition that the
straight lines may be at right angles is
a — ^gx + cx^-\- h — 2fy + cy'^+ 2 cos (o{h — gy —fx + ca;y) =0.
The centre of this circle
" (f •!)
Hence, whether the axes are rectangular or oblique the centre of the conic, which coincides with the centre of its
director-circle, is (-, - ) .
242. To find the foci of a conic whose tangential equation is given.
Let (f, Tj) and (f, i;') be a pair of foci (both being real or both imaginary). The product of the perpendiculars from these points on the line Ix + my +1=0 will be (?^+m7y + l)(^^^ + mV+l) I' + m^ This product will be equal to some constant X for all values of I and m if, V (ff' _ X) + im (^' + vS') +«i' (VV - X) + i (^ + f)
+ m (?; + t;') + 1 = 0. Comparing this with the tangential equation we have
a 2h ' b ' 2g 2/ c'"^ ^'
Hence cf f ' — ctjt}' = a—b, and c^rj' + CT/f ' = 2h. Eliminate ^' and rj' by means of the last two equations of (i), and we have
^{c^-2g)-v(cv-2f)^h-a, and ^{cy-2f)-\-r]{c^-2g) = -2L
Hence the foci are the four points of intersection of the two conies, cx^ — cy^ — 2gx + 2fy + a — 6 = 0, cxy —fx - gy+h = 0.
243. If >Si = 0 and S' = 0 be the tangential equations of two conies, then 8 — XS' = 0 will be the tangential equation
270 FOCI OF ENVELOPE.
of a conic touching the four common tangents of the first two.
For, if >Sf = 0 be
ar + ^hlm + hm^ + 2gl -f 2/m + c = 0, and>Sf' = Obe
aT + 2h'lm + hm' + 2g'l + 2/m + c' = 0.
Then S — \S' = 0 represeDts a conic, and any values of I and m which satisfy both S = 0 and S' = 0 will, what- ever \ may be, satisfy S — XS' =0. Therefore the conic S — XS' =0 will touch the common tangents of /S'= 0 and /Sf' = 0.
Ex. 1. The locus of the centres of all conies which touch four fixed straight lines is a straight line.
It S = 0, and S' = 0 be the tangential equations of any two conies which touch the four straight lines, S-\S'=0 w]ll be the general equation of the conies touching the lines. The centre of this conic is given by
Eliminating X we obtain the equation of the centre-locus, viz. « {cf - c'f) + y {c'g - eg') -f'g +fg' = 0.
Ex. 2. The director-circles of all conies which touch four straight lines have a common radical axis.
The director-circle of the conig S -\S' = 0 is
a + b-2gx- 2fij + c {x^ + y^)-\{a' + b' -2g'x- 2f'y + c' {x'' + y^)}^0.
[Art. 241.]
This circle always passes through the points common to the two circles
a:^ + y2_2^-x-2ly + ^-±^ = 0, c c' c
^ c' c '' c
The radical axis is therefore the line
<f-f;)-K^^.)-"-^
One of the conies of the system is a parabola, and its directrix is clearly the common radical axis of the director-circles.
EXAMPLES. 271
Examples on Chapter XII.
1. PN is the ordinate at any point P of a parabola whose vertex is -4, and the rectangle ANFM is completed; find the envelope of the line MN.
2. If the difference of the intercepts on the axes made by a moving line be constant, shew that the line will envelope a parabola.
3. Find the envelope of a straight line which cuts off a constant area from two fixed straight lines.
4. PiV, DM are the ordinates of an ellipse at the extremi- ties of a pair of conjugate diameters; find the envelope of PJD. Find also the envelope of the line through the middle points of ^^F and of Iin.
5. A£ and A'B' are two given finite straight lines, a line FF' cuts these lines so that the ratio AF : FJB is equal to A'F" : F'F'; shew that FF' envelopes a parabola which touches the given sti-aight lines.
6. OAF, OBQ are two fixed straight lines, A, B are fixed points and P, Q are such that rectangle AF . BQ is constant, shew that FQ envelopes a conic.
7. A series of circles are described each touching two given straight lines; shew that the polars of any given point with respect to the circles will envelop a parabola.
8. Two points are taken on an ellipse such that the sum of the ordinates is constant ; shew that the envelope of the line joining the points is a parabola.
9. A fixed tangent to a parabola is cut by any other tan- gent FT in the point T, and TQ is drawn perpendicular to TF; shew that TQ envelopes another parabola.
10. Through any point P on a given straight line a line FQ is drawn parallel to the polar of P with respect to a given conic; prove that the envelope of these lines is a parabola.
11. If a leaf of a book be folded so that one corner moves along an opposite side, the line of the crease will envelope a parabola.
12. An ellipse turns about its centre, find the envelope of the chords of intersection with the initial position.
272 EXAMPLES ON CHAPTER XII.
13. An angle of constant magnitude moves so that one side passes through a fixed point and its summit moves along a fixed straight line; shew that the other side envelopes a parabola.
14. The middle point of a chord FQ of an ellipse is on a given straight line; shew that the chord FQ envelopes a parabola.
15. 0 is any point on a conic and OF, OQ are chords drawn parallel to two fixed straight lines; shew that FQ envelopes a conic.
16. Any pair of conjugate diameters of an ellipse meets a fixed circle concentric with the ellipse in P, ^ ; shew that FQ will envelope a similar and similarly situated ellipse.
1 7. If the sum of the squares of the perpendiculars from any number of fixed points on a straight line be constant; shew that the line will envelope a conic.
18. The sides of a triangle, produced if necessary, are cut by a straight line in the points Z, J/, N respectively; shew that, if ZJf : MN be constant the line will envelope a parabola.
19. OA, OB are two fixed straight lines, and a circle which passes through 0 and through another g'^YQn fixed point cuts the lines in P, Q respectively ; shew that the line FQ en- velopes a parabola.
20. The four normals to an ellipse at P, Q, 7?, S meet in a point; prove that if the chord FQ pass through a fixed point, the chord RS will envelope a parabola.
21. A rectangular hyperbola is cut by a circle of any radius whose centre is at a fixed point on one of the axes of the hyperbola ; shew that the lines joining the points of intersection are either parallel to an axis of the hyperbola or are tangents to a fixed parabola.
22. Shew that the envelope of the polar of a given point with respect to a system of ellipses whose axes are given in magnitude and direction and whose centres are on a given straight line is a parabola.
23. Of two equal circles one is fixed and the other passes through a fixed point ; shew that their radical axis en- velopes a conic having the fixed point for focus.
EXAMPLES ON CHAPTER XII. 273
24. If pairs of radii vectors be drawn from the centre of an ellipse making with the major axis angles whose sum is a right angle, the locus of the poles of the chords joining their extremities is a concentric hyperbola, and the envelope of the chords is a rectangular hyperbola.
25. From any point on one of the equi- conjugate dia- meters of a conic lines are drawn to the extremities of an axis and these lines cut the curve again in the points P, Q ; shew that the envelope of PQ is a rectangular hyperbola.
26. FNF' is the double ordinate of an ellipse which is equi-distant from the centre C and a vertex ; shew that if parabolas be drawn through P, P', (7, the chords joining the other intersections of the parabola and ellipse will touch a second ellipse equal in all respects to the given one.
27. Two given parallel straight lines are cut in the points P, ^ by a line which passes through a fixed point ; find the envelope of the circle on PQ as diameter.
28. The envelope of the circles described on a system of parallel chords of a conic as diameters is another conic.
29. A chord of a parabola is such that the circle described on the chord as diameter will touch the curve ; shew that the chord envelopes another parabola.
30. Shew that the envelope of the directrices of all parabolas which have a common vertex A, and which pass through a fixed point P^ is a parabola the length of whose latus rectum is AP.
31. Prove that, if the bisectors of the internal and exter- nal angles between two tangents to a conic be parallel to two given diameters of the conic, the chord of contact will envelope an hyperbola whose asymptotes are the conjugates of those diameters.
32. The polar of a point P with respect to a given conic aS' meets two fixed straight lines AB^ ^C in Q^ Q' \ shew that, if -4 P bisect QQ\ the locus of P will be a conic; shew also that the envelope of QQ will be another conic.
S.C.S. 18
274 EXAMPLES ON CHAPTER XII.
33. If two points be taken on a conic so that the har- monic mean of their distances from one focus is constant, shew that the chord joining them will always touch a confocal conic.
34. The envelope of the chord of a parabola which sub- tends a right angle at the focus is the ellipse
(a;-3«)"+2/ = 8< i/-^ax=0 being the equation of the parabola.
35. A chord of a conic which subtends a constant angle at a given point on the curve envelopes a conic having double con- tact with the given conic.
36. Through a fixed point a pair of chords of a circle are drawn at right angles ; prove that each side of the quad- rilateral formed by joining their extremities envelope a conic of which the fixed point and the centre of the circle are foci.
37. The perpendicular from a point >S' on its polar with respect to a parabola meets the axis of the parabola in C; shew that chords of the parabola which subtend a right angle at S all touch a conic whose centre is C.
38. Shew that chords of a conic which subtend a right angle at a fixed point 0 envelope another conic.
Shew also that the point 0 is a focus of the envelope and that the directrix corresponding to 0 is the polar of 0 with respect to the original conic.
Shew that the envelopes corresponding to a system of con- centric similar and similarly situated conies are confocal.
39. A straight line meets one of a system of confocal conies in F, Q, and BS is the line joining the feet of the other two normals drawn from the point of intersection of the normals at P and Q. Prove that the envelope of EiS is a parabola touching the axes.
40. If a line cut two given circles so that the portions of the line intercepted by the circles are in a constant ratio, shew that it will envelope a conic, which will be a parabola if the ratio be one of equality.
41. Chords of a rectangular hyperbola at right angles to each other, subtend right angles at a fixed point O; shew that they intersect in the polar of*0.
EXAMPLES ON CHAPTER XII. 275
42. Shew that it AP, AQ be two chords of the parabola y^ — 4:ax = 0 through the vertex A, which make an angle
J with one another ; the line FQ will always touch the ellipse
I28a' I6a'
43. Pairs of points are taken on a conic, such that the lines joining them to a given point are equally inclined to a given straight line; prove that the chord joining any such pair of points envelopes a conic whose director- circle passes through the fixed point.
44. Chords of a conic S which subtend a right angle at a fixed point envelope a conic >S". Shew that, if jS pass through four fixed points, S^ will touch four fixed straight lines.
45. A conic passes through the four fixed points A, B, C, D and the tangents to it at B and C are met by CA, BA produced in P, Q. Shew that PQ envelopes a conic which touches BA, CA.
46. If a chord cut a circle in two points A, B which are such that the rectangle OA . OB is constant, 0 being a fixed point ; shew that the envelope of the chord is a conic of which 0 is a focus. Shew also that if OA^ + OB' be constant, the chord will envelope a parabola.
47. On a diameter of a circle two points A, A' are taken equally distant from the centre, and the lines joining any point P of the circle to these points cut the circle again in Q, P ; shew that QP envelopes a conic of which the given circle is the auxiliary circle.
48. A triangle is inscribed in an ellipse and two of its sides pass through fixed points ; shew that the envelope of the third side is a conic having double contact with the former.
49. A triangle is inscribed in an ellipse and two of its sides touch a coaxial ellipse ; shew that the envelope of the third side is a third ellipse.
50. Shew that the locus of the centre of a conic which is inscribed in a given triangle, and which has the sum of the squares of its axes constant, is a circle.
18—2
CHAPTER XIII. Trilinear Co-ordinates.
244. Let any three straight lines be taken which do not meet in a point, and let ABC be the triangle formed by them. Let the perpendicular distances of any point P from the sides BG, GA, ABhQ a, y8, 7 respectively; then a, /8, 7 are called the trilinear co-ordinates of the point P referred to the triangle ABG. We shall consider a, A 7 to be positive when drawn in the same direction as the perpendiculars on the sides from the opposite angular points of the triangle of reference.
Two of these perpendicular distances are sufficient to determine the position of any point, there must therefore be some relation connecting the three.
The relation is
aa -f 6y3 -}- C7 = 2 A, where A is the area of the triangle ABG. This is evidently true for any point P within the triangle, since the triangles BPG, GPA and APB are together equal to the triangle ABG ; and, regard being had to the signs of the perpendiculars, it can be easily seen to be universally true, by drawing figures for the different cases.
245. By means of the relation aa + hfi-\- cy — 2 A any equation can be made homogeneous in a, /Q, 7 ; and when we have done this we may use instead of the actual co- ordinates of a point, any quantities proportional to them : for if any values a, y8, 7 satisfy a homogeneous equation, then ka, k/S, by will also satisfy that equation.
TRILINEAR CO-ORDINATES. 277
246. If any origin be taken within the triangle, the. equations of the sides of the triangle referred to any rectangular axes through this point can be written in the form
— X cos 0^—y sin B^-\-p^ = 0,
— xcosd^—i/ sin 6^ + p^=0,
— X cos 0^ — y sin 0^-^ p^ = 0,
where cos {6^ — 6^- — cos A, cos (^3 — 6^ = — cos B, and cos {0^ — 6^ = — cos G.
[We write the equations with the constant terms posi- tive because the perpendiculars on the sides from a point within the triangle are all positive.]
We therefore have [Art. 31]
a=p^ — xcos6^ — ysinO^,
^=p^-xcos6^-y sin 6^,
and y=p^ — xcosd^ — y sin 0^.
By means of the above we can change any equation in trilinear co-ordinates into the corresponding equation in common (or Cartesian) co-ordinates.
247. Every equation of the first degree represents a straight line.
Let the equation be
loL + m^ -}- W7 = 0.
If we substitute the values found in the preceding Article for a, /3, 7, the equation in Cartesian co-ordinates so found will clearly be of the first degree. Therefore the locus is a straight line.
248. Every straight line can he represented by an equation of the first degree.
It will be sufficient to shew that we can always find values of I, m, n such that the equation la + m^ + ny= 0, which we know represents a straight line, is satisfied by the co-ordinates of any two points.
If the co-ordinates of the points be a, /3', 7' and
278
EQUATION OF A STRAIGHT LINE.
a", ^'\ 'y" we must have
W +my8' +ny =0,
and values of I, m, n can always be found to satisfy these two equations.
249. To find the equation of a straight line which passes through two given pohits.
Let a', yQ', 7'; a\ P", 7'' be the co-ordinates of the two points.
The equation of any straight line is
loL -}- m/3 + n<y = 0. The points (a, y8', 7), {a\ /3", 7"), are on the line if la! + mfi' + ny = 0, /a''+ ml3''+ ny" = 0. Eliminating Z, m, n from these three equations we have
a , P , 7
250. To find the condition that three given points may he on a straight line.
Let the co-ordinates of the given points be a, yS', 7' ;
a^/3^7"; and a'^r', 77.
If these are on the straight line whose equation is
lot -I- m^ + ny = 0,
we must have la.' +mfi' -f- ny = 0,
la' + m/S" + 7iy' = 0,
and la'" +mfi"' + ny'"=0.
Eliminating I, m, n we obtain the required condition, viz.
a' , ^' , 7
a > P > 7
lit r%i>t h
a , ^ * 7
0.
INTERSECTION OF STRAIGHT LINES.
270
251 . To find the point ofmtersection of two given straight lines.
Let the equations of the given straight lines be loL + mfi 4- n7 = 0, and I'a + m'^ + ?i 7 = 0.
At the point which is common to these, we have
^^=_^=._,x^ (i).
TYin — mn nl — nl Im — I'm ^
The above equations give the ratios of the co-ordinates. If the actual values be required, multiply the nume- rators and denominators of the fractions in (i) by a, b, c respectively, and add; then each fraction is equal to aoL + hl3+ cy 2 A
1 • Q-£
a{mn — mn) + h {nl' — nl) -f- c {Im' — I'm) '
|
I, m, |
n |
|
l\ m\ |
n |
|
a, 6, |
c |
The lines will not meet in a point at a finite distance from the triangle of reference, that is to say the lines will be parallel, if
Z, m, n = 0.
r, m, n
a, b, c
252. To find the condition that three straight lines max) Tneet in a point.
Let the equations of the straight lines be \i+m.fi + n^y = 0, l^a + mfi + n^ = 0, Zga-f-mg^ 4- 7137 = 0. The lines will meet in a point if the above equations are all satisfied by the same values of a, /3, 7. The elimi- nation of a, yS, 7 gives for the required condition Zj, m., n. =0. ky 'f^K, n„
I
"2'
280 LINE AT INFINITY.
253. If Ax + Bt/ + (7= 0 be the equation of a straight line in Cartesian co-ordinates, the intercepts which the
line makes on the axes are — j , — -» respectively. If
therefore A and B be very small the line will be at a very great distance from the origin. The equation of the line will in the limit, assume the form
0.x-\-0.y-\-C=0. The equation of an infinitely distant straight line, generally called the line at infinity, is therefore 0.a; + 0.y + C=0. When the line at infinity is to be combined with other expressions involving x and y it is written C=0.
The equation of the line at infinity in trilinear co-ordi- nates is aa + h^ + cy = 0.
For if ka, Jeff, ky be the co-ordinates of any point, the invariable relation gives k{aa+ hff + cy) = 2 A, or
2A
If therefore h become infinitely great, we have in the limit the relation aa + 6y9 + C7 = 0. This is a linear relation which is satisfied by finite quantities which are propor- tional to the co-ordinates of any infinitely distant point, and it is not satisfied by the co-ordinates, or by quantities proportional to the co-ordinates, of any point at a finite distance from the triangle of reference.
254. To find the condition that two given lines may he parallel.
Let the equations of the lines be loL + myS â– \-ny = 0, Vol -}- mff 4- ny = 0. If the lines are parallel their point of intersection will be at an infinite distance from the origin and therefore its co-ordinates will satisfy the relation
aa + 6/5 + C7 = 0.
PARALLEL LINES. 281
Eliminating a, /S, 7 from the three equations, we have the required condition, viz.
I , m, n =0.
If r t
I y m , n a, b , c
255. To find the equation of a straight line through a given point parallel to a given straight line.
Let the equation of the given line be
loL 4- w/S + 717 = 0. The required line meets this where
aa + 6^ + C7 = 0. The equation is therefore of the form
Za + wyS + 717 + X (aa + 6;8 + 07) = 0.
If /, g, h be the co-ordinates of the given point, we must also have
Z/*+ mg + nh + X {af+ hg + ch) = 0, , ?a + m/3 4- «7 _a2-\-h^ + cy
If + mg + nh af+bg + ch'
A useful case is to find the equation of a straight line through an angular point of the triangle of reference parallel to a given straight line.
If A be the angular point, its co-ordinates are f 0, 0, and the equation becomes {ma —lh)p-\- (na — lc)y — 0.
256. To find the condition of perpendicidarity of two given straight lines.
Let the equations of the lines be loL + m^ 4- ^7 = 0, Vol + m'yS + ny = 0.
If these be expressed in Cartesian co-ordinates by- means of the equations found iu Art. 246, they will be x{l cos 6^-\-m cos 6^-{- n cos ^3) + y{l sin ^j+ msin 6^-\-n^\n6^
â– -b\-'mp^-np^ = 0,
282 LENGTH OF A PERPENDICULAR.
and
X (r cos ^j + m cos ^2 + ^' cos ^3) + y {V sin 6^ + m sin 6^ + n' sin ^3)
the lines will therefore be perpendicular [Art. 29] if
{I cos ^j + m cos 6^-\-n cos ^3) (^ cos 6^ + 7?i' cos ^^ 4- n cos ^3) + (Z sin ^j + m sin 6^+n sin ^g) (Z' sin 6^ + m' sin 6^ + n' sin ^3) = 0 ; that is, if
IV + mm + w?i' + (Zwi' + I'm) cos (^^ ~ ^2)
+ {mn + m'w) cos {d^ ~ ^3) + {nl + /I'Z) cos (^3- B^ = 0. But cos (^2 — ^3) = — cos A, cos (^3 — 6^ = — cos ^, and cos (6^ -^0^ = — cos (7 ;
therefore the required condition is
IV + mm' + nn — (mn 4- m'?i) cos A — {nV + nl) cos B
-{Im +rm) cos C=0.
257. To find the perpendicular distance of a given point from a given straight line.
Let the equation of the straight line be loL + m^ -j-ny = 0. Expressed in Cartesian co-ordinates the equation will be X (l cos ^j + m cos 0^+n cos ^3) +y{l sin ^^ + m sin 0^+n sin ^3)
-lp^-mp^-n2\ = 0. The perpendicular distance of any point from this line is found by substituting the co-ordinates of the point in the expression on the left of the equation and dividing by the square root of the sum of the squares of the coefficients of X and y. If this be again expressed in trilinear co- ordinates, we shall have, for the length of the perpen- dicular from f g, h on the given line, the value
lf+ mg + nh ^/{(lcos6^-{-mcos6^+ncosdJ^-\'{lsmd^-\-msmd^+nsmOJ^] '
The denominator is the square root of P + m^+ n"- -}- 2Zm cos {d^ - d^) -t- %nn cos [6^ - 6^)
-\-2nl cos (0^-6^), or of r + 7n^ + n^ — 2lm cos C — 2mn cos A — 2nl cos B.
CO-ORDINATES OF FOUR POINTS. 283
Hence the length of the perpendicular is equal to
lf+ mg + nh VOT^ + m^ + ri^ — 2mn cos A — 2nl cos B — 2lm cos C) '
258. To shew that the co-ordinates of any four points may he expressed in the form, ±f, ±g, ±h.
Let F, Q, R, S be the four points.
B
The intersection of the line joining two of the points and the line joining the other two is called a diagonal- pohit of the quadrangle. There are therefore three diagonal-points, viz. the points A, B, (7 in the figure.
Take ABC for the triangle of reference, and let the co-ordinates of P be/, g, h.
Then the equation oi AP will be - =4- .
The pencil AB, AS, AG, AP is harmonic [Art. 60], and the equations oi AB, A C are y = 0, ^ = 0 respectively,
and the equation of AP is - = 7^ : therefore the equation
g h
o£AS will be ^ = -^ . [Art. 56.1 g -h ^
(1 R The equation of GP is 7. = — .
Therefore where AS and GP meet, i.e. at /S', we shall
have ?=^= 'y.
7 9 -h'
284
EQUATIONS OF FOUR STRAIGHT LINES.
So that the co-ordinates of S are proportional to f, g, — h. Similarly the co-ordinates of jR are proportional to —f,g, h. Similarly the co-ordinates of Q are proportional to /, — g, h.
259. To shew that the equations of any four straight lines may be expressed in the form 1% ± m/3 + n^ = 0.
Let DEF, DKG, EKE, FGH be the four straight lines.
Let ABC be the triangle formed by the diagonals FK, EG, and DZTof the quadrilateral, and take ABC for the triangle of reference.
Let the equation of DEF be
loL 4- m^ + ^7 = 0. Then the equation of -4 Z) is m/3 + ^7 = 0.
Since the pencil AD, AB, AH, -4(7 is harmonic [Art. 60], and the equations oiAD, AB, AG are m/9-t-?i7=0, 7 = 0, y8 = 0 respectively ; therefore [Art. 56] the equation of AHi^ m^ — ny = 0.
Since ^ is the point given by )S = 0, la.+ ny = 0 ; and H is the point given by a=0, m/3 — ^17 = 0 ; the equation of HE is h-m/3 + ?iy = 0.
EXAMPLES. 285
We can shew in a similar manner that the equation of DK is - Za + my8 + ny = 0,
and that the equation of FH is
la + m^ — ny = 0.
EXAMPLES.
1. The three bisectors of the angles of the triangle of reference have for equations, /3-7=0, 7-a=0, and a-j3 = 0.
2. The three straight lines from the angular points of the triangle of reference to the middle points of the opposite sides have for equations b^-cy = 0, cy-aa = 0, and aa-&/3=0.
3. If A'B'C be the middle points of the sides of the triangle of reference, the equations of B'C, G'A', A'B' will be &/3 + C7-aa = 0, C7 + aa - 6^ = 0, aa + h^-cy = 0 respectively.
4. The equation of the line joining the centres of the inscribed and circumscribed circles of a triangle is
a (cos B - cos C) + /S (cos G - cos A) + y (cos A - cos B) = 0.
5. Find the co-ordinates of the centres of the four circles which touch the sides of the triangle of reference. Find also the co-ordinates of the six middle points of the lines joining the four centres, and shew that the co-ordinates of these six points all satisfy the equation
6. If AO, BO, CO meet the sides of the triangle ABC in A', B\ C'\ and if B'C meet BC in P, C'A' meet CA in (?, and A'B' meet ABinB-, shew that P, Q, 12 are on a straight line.
Shew also that BQ, CR, AA' meet in a point P'; CR, AP, BB' meet in a point Q'; and that AP, BQ, CC meet in a point R'.
7. If through the middle points, A\ B', C of the sides of the triangle ABC lines A'P, B'Q, CR be drawn perpendicular to the sides and equal to them; shew that AP, BQ, CR will meet in a point.
8. If p, q, r be the lengths of the perpendiculars from the angular points of the triangle of reference on any straight line; shew that the equation of the line will be apa + bqp + cry=0.
9. If there be two triangles such that the straight lines joining the corresponding angles meet in a point, then will the three intersections of corresponding sides lie on a straight line.
[Let/, g, h be the co-ordinates of the point, referred to ABC one of the two triangles. Then the co-ordinates of the angular points of the other triangle A'BC can be taken to be/', g, h ; /, g', h and/, g, h' respectively.
S y
B'C cuts BC where a = 0 and -^-—, + , \, = 0. Hence the three infcer-
g-g' h-h'
sections of corresponding sides are on the line -; — r, H — ~-, + , -77 = 0-1
/--./ g~g h-h' ^
286 TANGENT AND POLAR.
260. The general equation of the second degree in trilinear co-ordinates, viz.
u^ + v^'' 4- w^'' + 2z^'^7 + 2?;'7a + 'Iw'a.fi = 0,
is the equation of a conic section ; for, if the equation be expressed in Cartesian co-ordinates the equation will be of the second degree.
Also, since the equation contains five independent constants, these can be so determined that the curve represented by the equation will pass through five given points, and therefore will coincide with any given conic.
261. To find the equation of the tangent at any point of a conic.
Let the equation of the conic be </) (a, y9, 7) = wa' + vjB^ + wrf + 2w'y87 + 2vrya + 2wa^ = 0, and let a!, ^', 7' ; a", ^'\ 7" be the co-ordinates of two points on it.
The equation
u (a-a') (a- a') + v (0-13') {^-fi") + w(y-ry')(y-y") + 2u' (^ - fi') (7 - 7") + 2v' (7 - 7O (a - a")
is really of the first degree in a, /8, 7, and therefore it is the equation of some straight line. The equation is satisfied by the values a = a', /3 = /3', 7 = 7', and also by the values a = a", y3=^", 7 = 7". Therefore it is the equation of the line joining the two points (a, ^\ 7'), (a", /3", 7"). Let now {a\ ^", 7") move up to and ulti- mately coincide with (a', ^, 7'), and we have the equation of the tangent at (a', yS', 7'), viz.,
uaa! + v^^' + ^77 + n {M + 7/^')
+ v' {yx + o^y') + w' {a/S' + ^ol) = 0.
Using the notation of the Differential Calculus we may write the equation of the tangent at any point {a, /3', 7')
CONDITION OF TANGENCY.
287
of the conic <^ (a, A 7) = ^ i^ either of the forms
0,
or
d<f> ^ d6 d(f)
262. To find the equation of the polar of a given point.
It may be shewn, exactly as in Art. 76, 100, or 118, that the equation of the polar of a point with respect to a conic is of the same form as the equation we have found for the tangent in Art. 261.
263. To find the condition that a given straight line may touch a conic.
Let the equation of the given straight line be
l'x+m^-\-ny = 0 (i).
The equation of the tangent at (a', p', y) is a (uoi' + w'^' + v'y) + ^ {w'a! + v^' + uy)
+ 7 \v'a' + u'jS' + wy) = 0... (ii). If (i) and (ii) represent the same straight line, we have ua' + w'^' 4- vy _ w'ol + vfi' + u'y _ v(j! + u^' + wy
I m n '
Putting each of these fractions equal to — X, we have wjL + w^' + vy -rXl =0, w'a+vfi' +uy+\m = 0, va + u^' + wy +\n =0. Also, since (a', /3', 7') is on the line {I, m, n),
la + m^' + ny = 0. Eliminating ol, /3', 7', X from these four equations we obtain the required condition
(iiij,
|
u , |
w\ |
v\ |
I |
= 0 |
|
w\ |
V , |
u\ |
m |
|
|
v' , |
11, |
lU, |
n |
|
|
I , |
m, |
n, |
0 |
288
THE CENTRE.
or r . {vw —u'^) + m^ {wu — v^) -\- n^ [uv — w'^)
+ 2mw [v'w' — uu") + 2nl {w'u' — vv') + 2lm {uv — ww) = 0, or
UP + Vm^ + Wn' + 2 U'mn + 2 Vnl + 2 F7m = 0. . . (v), where U, F, TT, CT, V\ W are the minors of u, v, w, u, v\ w' in the determinant
0.
|
u , |
w, |
V |
|
w\ |
V , |
u |
|
v\ |
< |
w |
264. To find the co-ordinates of the centre of a conic. Since the polar of the centre of a conic is altogether at
an infinite distance, its equation is
aa + 6/3 + C7 = 0 (i).
But [Art. 262], the equation of the polar of the centre will be
da^ "^ d/3, dy^ where o^, yS^, 7^, are the co-ordinates of the centre. Hence the equations for finding the centre are d(f> d(f> d<f)
a h c '
265. To find the condition that the curve i^epresented hy the general equation of the second degree may he a parabola.
The co-ordinates of the centre of the curve are given by the equations
a h c '
Put each of these equal to - X, and we have ua^ + w'^^ -f vy^ + Xa = 0, w'a^-hv/3^ +W7j, + X6 = 0, va^ + u'^^ + rvy^ + \c = 0.
ASYMPTOTES.
289
Also since the centre of a parabola is at infinity, we have
The elimination of a„, yS^, 7o> ^ gives for the required condition
u y w\ V , a =0. w\ V , u, h V' , 11 , w , c a , h , c, 0 We see from the above that the parabola touches the line at infinity. [Art. 263.]
266. To find the condition that the conic represented hy the general equation of the second degree may he two straight lines.
The required condition may be found as in Art. 37. The condition is
uvw + 2uvw' — uii^ — vv or, as a determinant,
w
W, V ,
= 0.
267. To find the asymptotes of a conic.
The equations of the curve and of its asymptotes only differ by a constant.
Hence if the equation of the curve be
uoL^+ v/S^ + wy^ + 2ul3y + 2vyoL + 2io'a^ = 0, the equation of the asymptotes will be
wa' + v^ + wy^ + 2u'l3y + 2v'ya + 2wa^
+ \(aa + 6y8 + C7)'=0 (i). ^
The value of X is to be determined from the condition for straight lines, viz.
u + \o^ , w + Xa6,
w' -\-\oh, v' -1- \aCy
11 + \hc,
V 4- "y^ac u'+ \hc w +\c^
= 0.
s. c. s.
19
290 ASYMPTOTES.
The term independent of X is
|
u |
, w |
, -y |
||||||
|
w |
, V |
,u' |
. |
|||||
|
v' |
.u |
, -m; |
||||||
|
The coefficient of \ is |
||||||||
|
a% ah, ac |
+ |
u |
, w |
, ^' |
+ |
M , |
w. |
|
|
w\ V , u |
ah |
, h^ |
, he |
w, |
V , |
|||
|
V , u , w |
v' |
, -m; |
ac, |
he, |
||||
|
ich is equal to |
||||||||
|
— |
u , |
w , |
V , a |
|||||
|
w\ |
V , |
xC , h |
||||||
|
v\ |
u , |
w , c |
||||||
|
1 |
a , |
h , |
c , |
ol |
The coefficients of \^ and of V are both zero.
Hence there is a simple equation for \, and therefore from (i) we have for the equation of the asymptotes <^(a,ft7) u , w , V , a -{-{aoL^-h^^-cyf u , w , v' = 0.
|
u , w' , V |
a |
+(aoL-\-h^-\-cyf |
U y w , v' |
|
w\ V , u |
, h |
v/ 1 V , u' |
|
|
V , u\ w |
, c |
V J u' , w |
|
|
a , h , c |
, 0 |
268. To find the condition that the conic may he a rectangular hyperhola.
Change to Cartesian co-ordinates. Then the conic will be a rectangular hyperbola if the sum of the coefficients of a? and y"^ is zero.
The condition becomes u + v + w — 2u cos A — 2u' cos B — ^w cos (7=0.
269. To find the equation of the circle circumsciHhing the triangle of reference.
If from any point P, on the circle circumscribing a triangle ABC, the three perpendiculars PL, PM, PN be drawn to the sides of the triangle and meet the sides
CIRCUMSCRIBING CIRCLE. 291
BG, GAy AB in the points L, M, ^respectively; then it is known that these three points L, M, N are in a straight line.
Let the triangle be taken for the triangle of reference and let a, A 7 be the co-ordinates of P.
The areas of the triangles MPN, NFL, and LPM are ^fiysmA, J 7a sin jB, and JaySsinO respectively. Since X, M, N are on a straight line, one of these triangles is equal to the sum of the other two. Hence, regard being had to sign, we have
^7 sin A + yoLsmB-{- a/3 sin G = 0, or ajSy + hyoL + ca/3 = 0,
which is the equation required.
Ex. The perpendiculars from O on the sides of a triangle meet the sides in D, E, F. Shew that, if the area of the triangle DEF is constant, the locus of 0 is a circle concentric with the circumscribing circle.
270. Since the terms of the second degree are the same in the equations of all circles, if S = 0 be the equation of any one circle, the equation of any other circle can be written in the form
or, in the homogeneous form,
8+{la + mff + ny) (aa+h^ + cy) = 0.
271. From the form of the general equation of a circle in Art. 270 it is evident that the line at infinity cuts all circles in the same two (imaginary) points.
The two points at infinity through which all circles pass are called the circular points at infinity.
Since, in Cartesian co-ordinates, the lines x^ + y^ = 0 are parallel to the asymptotes of any circle, the imaginary lines x'^-{-y^ = 0 go through the circular points at infinity. Hence, from Art. 193, the four points of intersection of the tangents drawn to a conic from the circular points at infinity are the four foci of the curve.
272. To find the conditions that the curve represented hy the general equation of the second degree may he a circle.
19—2
292 CONDITIONS FOR A CIRCLE.
The equation of the circle circumscribing the triangle of reference is [Art. 269]
a^y + 67a + ca/3 = 0. Therefore [Art. 270] the equation of any other circle is of the form
ay87 + 67a + ca/3 + (^ + m^ + 717) (aa + 6;8 + 07) = 0. If this is the same curve as that represented by ibd^ + v^^ + wrj" + 2i*'/37 + 2^; 7a + ^w'a^ = 0, we must have, for some value of X,
\u = lay Xv = mhy Xw = nc, 2\u' — a + cm + hn, 2\v' = 6 + a?i + c^,and 2\w' = c 4- 6^ + am. Hence 2hcu' — cV — 6'^z^? = 2cav' — a^w — c^u = 2ahu' — h\ — a\
for each of these quantities is equal to — - .
273. To find the condition that the conic represented hy the general equation of the second degree may he an ellipse, pai^ahola, or hyperbola.
The equation of the lines from the angular point G to the points at infinity on the conic will be found by elimi- nating 7 from the equation of the curve and the equation ax + Z>y3 + C7 = 0. Hence the equation of the lines through 0 parallel to the asymptotes of the conic will be
wc'a' + vc'yS' ^w{aoL + b/Sy - 2ucff (aa + b/S)
- 2vca (aa + b^) + 2w c'ayS = 0. The conic is an ellipse, parabola, or hyperbola, accord- ing as these lines are imaginary, coincident, or real ; and the lines are imaginary, coincident, or real according as (wab — uac — vbc + w'c^y — (uc^ + wa^ — 2v ac)
{vc' + w¥-2ubc) is negative, zero, or positive ; that is, according as Ua'+ Vb'-h Wc'-{-2U'bc + 2rca-h2W'ab is positive, zero, or negative.
274. The equation of a pair of tangents drawn to the conic from any point can be found by the method of Art. 188.
CIRCUMSCRIBING CONIC. 293
The equation of the director circle of the conic can be found by the method of Art. 189.
The equation giving the foci can be found by the method of Art. 193.
The equations for the foci will be found to be
4 (h'w + &v - 2bcu') (/, (a, ^^y)-(b^-c ^J = 4 {c'u + a'w - 2cav') ^ {a, ^,y)-(c^- a^J
The elimination of </> (a, /8, 7) will give the equation of the axes of the conic.
275. To find the equation of a conic circumscribing the triangle of reference.
The general equation of a conic is
ua^ + vff^ + wy^ + 2ul3y + 2vyoL + 2wal3 = 0. The co-ordinates of tKe angular points of the triangle are
— , 0, 0; 0, ^, 0 ; and 0, 0, — . a b c
If these points are on the curve, we must have u = 0,v=0, and w = 0, as is at once seen by substitution.
Hence the equation of a conic circumscribing the tri- angle of reference is
ufiy + v'yoL + w'oL^ = 0.
276. The condition that a given straight line may touch the conic may be found as in Art. 263, or as follows.
The equation of the lines joining A to the points common to the conic and the straight line (l, m, n), found by eliminating a between the equations of the conic and of the straight line, is
lu/3y - {vy + w'l3) (ml3 + 717) = 0, or mw'fi^ + nv'y^ + (mv' + nw' — lu) ^y = 0.
The lines are coincident if (l, m, n) is a tangent; the condition for this is
4imnv'w' — [mv + nw' — lu'Y^
294 INSCRIBED CONIC.
which is equivalent to
JTv! + Jmv ± Jnw = 0.
277. To find the equation of a conic touching the sides of the triangle of reference.
The general equation of a conic is
ua.^ + v^^ + wr/" + 2u^y + 2v'7a + 2w a/3 = 0. Where the conic cuts a = 0, we have ?)/3'+w;7'+2w'^7 = 0. Hence, if the conic cut a = 0 in coincident points we
have
t)w = u'^y or u' — Jvw.
Similarly, if the conic touch the other sides of the triangle, we have
v' = Jwu, and w — Juv. Putting X^, ^, v^ instead of u^ v, w respectively, we have for the equation,
W + fM^^^ + vy T 2fjLv^y + 2i/X7a + 2 Va/3 = 0. In this equation either one or three of the ambiguous signs must be negative ; for otherwise the left side of the equation would be a perfect square, in which case the conic would be two coincident straight lines. The equation can be written in the form s/M-\-Jfil3 + J^ = 0.
278. To find the condition that the line la + m^ + ny = 0 may touch the conic Jxa + J/jl^ + Jvy = 0.
The condition of tangency can be found as in Art. 276, the result is
L m n
279. To find the equations of the circles which touch the sides of the triangle of reference.
If D be the point where the inscribed circle touches BG, we know that
i>(7= 5 - c, and i)5 = 5 - J.
CONICS THROUGH FOUR POINTS. 295
Therefore the equation of AD will be ^ 7
(s-c) sin G^ {s-b)smB ••^'^•
Now the equation of any inscribed conic is
^/x^+y/i^ + ^/i^ = 0 (ii).
The equation of the line joining A to the point of contact of the conic with JSG will be given by
r. fi0 = vy (iii).
Hence, if (ii) is the inscribed circle, we have from (i) and
(iii)
fJL _ V
h{s — b)~c{s — c)' Similarly, by considering the point of contact with GA, we have
c{s — c) a(s — a)' Hence the equation of the inscribed circle is
Ja {s - a) a-^ Jb {s - b) 13 -\- J c (s - c) y = 0. The equations of the escribed circles can be found in a similar manner.
Ex. 1. Shew that the conic whose equation is Jaa + Jb^ + Jcy = 0, touches the sides of the triangle of reference at their middle points.
Ex. 2. If a conic be inscribed in a triangle, the lines joining the angular points of the triangle to the points of contact with the opposite sides will meet in a point.
280. To find the equation of a conic which passes through four given points.
If the diagonal-points of the quadrangle be the angular points of the triangle of reference, the co-ordinates of the four points are given by ±f ±g, ±h [Art. 258].
If the four points are on the conic whose equation is uoi" + vl3'' + wy'' + 2ii^y + 2v'ya + 2w'a0 = 0,
296 CONICS TOUCHING FOUR LINES.
we have the equations
XI f + vg" + wW- ± 2w'^A ± 2v V ± ^'^'fg = 0 ; .-. u' = v' = 10' — 0. Hence the equation of the conic is ua^ + v^'^ + wy"^ = 0, with the condition uf^ + vg^ + wh^ = 0.
Ex. 1. Find the locus of tJie centres of all conies which pass through four given points.
Let the four points be ±/, ±g, ±^. The equation of any conic will be
uu^ + v^+wy^^O, with the condition
iif^ + vg^ + wh^ = 0 (i).
The co-ordinates of the centre of the conic are given by Ma _ Vj8 _ wy a ~ h ~ c ' Substitute for m, v, to in (i), and we have the equation of the locus, viz.
-/^^%_^1^0. [See Art. 209.] a ^ 7
Ex. 2. The polars of a given point with respect to a system of
conies passing through four given points will pass through a fixed point.
281. To find the equation of a conic touching four give7i straight liifies.
Let the triangle formed by the diagonals of the quadri- lateral be taken for the triangle of reference, then [Art 259] the equations of the four lines will be of the form loL ± m^ ±ny = 0. The conic whose equation is uoc" + v^^ + wrf + 2w'^7 + 2vVa + ^w'a^ = 0 . . . (i) will touch the line (Z, m, n) if
m'' + Ym'' + F?i' + 2 TJ'mn + 2 Tnl + 2 Wlm = 0. If therefore the conic touch all four of the lines, we must have U'=V=W'^ 0,
that is v'w' — uu = 0,
w'u' — vv' = 0; uv — wiu' = 0 ; .'. u -^v =w' = 0,
SELF-POLAR TRIANGLE. 297
or else (i) is a perfect square, and the conic a pair of coincident straight lines.
Hence we must have u' =v' = w = 0, and the condition of tangency is l^w + m^wu + nhuv = 0.
Hence every conic touching the four straight lines is included in the equation
ud' + vfi'^ + wrf^O, with the condition
- + — + - = 0. U V w
Ex. 1. Find, the locus of the centres of the conies which touch four given straight lines.
Any conic is given by
ua^ + vp^ + wy^ = 0, with the condition
- + — + - = 0. U V w
The co-ordinates of the centre of the conic are given by wa _ v/S _ wy ^ a b ~ c ' therefore the equation of the locus of the centres is the straight line
ah c
This straight line goes through the middle points of the three diagonals of the quadrilateral. [See Art. 217.]
Ex. 2. The locus of the pole of a given line with respect to a system of conies inscribed in the same quadrilateral is a straight line.
Ex. 3. Shew that, if the conic ua'^+v^'^ + wy^ — Q be a parabola, it will touch the four lines given by aa±6^±C7 = 0.
282. When the equation of a conic is of the form ud^ + v^"^ 4- wr/ = 0, each angular point of the triangle of reference is the pole of the opposite side. This is at once seen if the co-ordinates of an angular point of the triangle be substituted in the equation of the polar of (a', /9', 7'), viz. UOLOL + v^'^ + W7V = 0.
Conversely, if the triangle of reference be self-polar, the equation of the conic will be of the form ucl^ + v^^ -|- wy^ = 0.
For, the equation of the polar oi A (■— , 0, 0 J , with respect
298 NINE-POINT CIRCLE.
to the conic given by the general equation, is
ua + w'p + vy = 0. Hence, if BG he the polar of A, we have lo' =v =0. Similarly, if CA be the polar of B, we have w =u' = 0. Hence w', v\ w' are all zero.
283. If two conies intersect in four real points, and we take the diagonal-points of the quadrangle formed by the four points for the triangle of reference, the equations of the two conies will [Art. 280] be of the form
un^ + v^'^ -\- w^ = 0, and wV -h v^^ 4- w'y^ = 0. So that, as we have seen in Art. 213, any two conies which intersect in four real points have a common self-polar triangle.
When two of the points of intersection of two conies are real and the other two imaginary, two angular points of the common self-polar triangle are imaginary. When all four points of intersection of two conies are imaginary, they will have a real self-polar triangle. [See Ferrers' Trilinears, or Salmon's Conic Sections, Art. 282.]
284. If two tangents and. their chord of contact be the sides of the triangle of reference, the equation of the conic will be of the form
a' = 2A./37 (i),
for (i) is a conic which has double contact with the conic ^y = 0, the chord of contact being a = 0. [See Art. 187.]
285. To find the equation of the circle with respect to which the triangle of reference is self -polar.
The equations of all conies with respect to which the triangle of reference is self-polar are of the form
The equation of any circle can be written in the form ajSy -\- hyoL H- ca^ -H (Xa + ^i^ + vy) {aa + b^ + cy) = 0. If these equations represent the same curve, we have
u = \a, v = fMh, w — vc, a-r fic + vb= 0, b -\- va + \c = 0, and c + \b + /jLa = 0.
pascal's theorem. 299
Whence X = — cos A, fM = — cos B, and v = — cos G. The required equation is therefore
a cos J. . a^ + 6 cos J5 . ^^ + c cos (7 . y'' — 0.
286. To find the equation of the nine-point circle. Let the equation of the circle be
a/37 -f hyoL + ca/5 - (Xa + /^/S + 1/7) (aa + &/5 + C7) = 0. This circle cuts a = 0 where h^ = cy ; .'. abc — 2 (fjLC + vh) he = 0,
h G 2ahc ' Similarly l + h £bc'
and - + f = ^TT •
a 0 zabc
Hence 2\ = cos A, 2/jl = cos B, and 2z/ = cos C ; therefore the equation of the circle is 2a^7 + 267a + 2coil3
- {(X cos A + 13 cos B + y cos G) (aoL + &/3 + cy) = 0, or a/87 + ^« + cayS — a^a cos J. — /8^6 cos B — y^c cos (7=0. The form of this equation shews that the nine-point circle, the circumscribed circle, and the self-conjugate circle have a common radical axis, the equation of the radical axis being
a cos A-\-fi cos -B -f 7 cos (7 = 0. Ex. 1. The centre of the self-conjugate circle of a triangle is its orthocentre.
Ex. 2. The locus of the centres of all rectangular hyperbolas de- scribed about a given triangle is the nine-point circle.
287. Pascal's Theorem. If a hexagon he inscribed in a conic, the three points of intersection of the three pairs of opposite sides will he on a straight line.
Let the angular points of the hexagon be A,F,B,D,C,E. Take ABG for the triangle of reference, and let the points D, E, J- be («'. /S; y), (a", &', 7"), and (a'", /3"', V")-
:300
PASCAL S THEOREM.
Let the equation of the conic be
^ + ^ + "^ = 0
(i).
The equations of BD and AE will be - = -, , and ^ a 7
-^ = -V, ; therefore at their intersection, a /3" 1 * A 7 - —
7 ry
Similarly OD, u4jP meet in the point i-^,, 1, XrJ •
1,^, -ITf]'
The three points will lie on a straight line if
|
a' |
/8" |
1 |
= 0, or if |
|
7 |
7 |
III |
|
|
a |
1, |
y |
|
|
/8^' |
/3"' |
||
|
1, |
^" |
III y |
|
|
!?'> |
i/i |
||
|
OL |
a |
|
1 |
1 |
1 |
|
7 |
7" |
7" |
|
1 |
1 |
1 |
|
)S" |
r' |
^"' |
|
1 |
1 |
1 |
|
a" |
a''' |
a''' |
0 (ii).
But, since the three points D, E, F are on the conic (i), we have
a' + ^ ^ y ' '
land
By the elimination of X, /x, i^ we see that the condition (ii) is satisfied, which proves the proposition. [See also Art. 319, Ex. 3.]
Since six points can be taken in order in sixty different ways, there are sixty hexagons corresponding to six points on a conic ; and, since Pascal's Theorem is true for every
brianchon's theorem. 301
one of these hexagons, there are sixty Pascal lines corre- sponding to six points on a conic.
288. If a hexagon circumscribe a conic, the points of contact of its sides will be the angular points of a hexagon inscribed in the conic. Each angular point of the circum- scribed hexagon will be the pole of the corresponding side of the inscribed hexagon ; therefore a diagonal of the cir- cumscribing hexagon, that is a line joining a pair of its opposite angular points, will be the polar of the point of intersection of a pair of opposite sides of the inscribed hexagon. But the three points of intersection of pairs of opposite sides of the inscribed hexagon lie on a straight line by Pascal's Theorem ; hence their three polars, that is the three diagonals of the circumscribing hexagon, will meet in a point. This proves Brianchon's Theorem : — if a hexagon he described about a conic, the three diagonals will meet in a /point.
289. If we are given five tangents to a conic we can find their points of contact by Brianchon's Theorem. For, let A, B, G, D, E be the angular points of a pentagon formed by the five given tangents; then, if Z'be the point of contact of AB, A, K, B, G, D, E are the angular points of a circumscribing hexagon, two sides of which are co- incident. By Brianchon's Theorem, DK passes through the point of intersection of ^0 and BE-, hence K is found. The other points of contact can be found in a similar manner.
Similarly, by means of Pascal's Theorem, we can find the tangents to a conic at five given points. For, let A, By G, i), E be the five given points, and let F be the point on the conic indefinitely near to A ; then, by Pascal's Theorem, the three points of intersection of AB and DE; oi BG and EF-, and of GD and FA lie on a straight line. Hence, if the line joining the point of intersection of AB and DE to the point of intersection oi BG and EA meet GD in Hy AH will be the tangent at A. The other tangents can be found in a similar manner.
302 AREAL CO-ORDINATES.
AREAL CO-ORDINATES.
290. The position of any point P is determined if the ratios of the triangles PBG, PGA, PAB to the triangle of reference ABG be given. These ratios are denoted by x, y, z respectively, and are called the areal co-ordinates of the point P.
The areal co-ordinates of any point are connected by the relation x + y + z = l.
Since a; = ^rr > y=J^> and z = ^ , we at once find 2A "^ 2A 2A
the equation in areal co-ordinates which corresponds to any
given homogeneous equation in trilinear co-ordinates, by
OS 7J z
substituting in the given equation - , r, - for a, yS, 7
respectively ; for example the equation of the line at in- finity is x + y-\-z = 0. We will however find the areal equation of the circumscribing circle independently.
291. To find the equation in areal co-ordinates of the circle luhich circwnscribes the triangle of reference.
If P be any point on the circle circumscribing the tri- angle ABG, then by Ptolemy's Theorem (Euclid VI. D.) we have
PA.BG±PB.GA±PG.AB=0 (i).
But since the angles BPG and BA G are equal, we have
PB PG
. P ' A^ = CO, and similarly for y and z; hence, paying
regard to the signs of x, y, z, we have from (i)
PA.PB.PG , PA.PB.PG PA.PB.PG ^
a , -, h 0 . 1- c . j = 0,
OCX cay aoz
<i"h'c' „ or — h -- -I- - = 0,
X y z
which is the equation required.
TANGENTIAL CO-ORDINATES. 303
292. If the conic represented by the general equation of the second degree in trilinear co-ordinates, viz.
ua^ + v^ + w^ -F ^u^^ 4- 2v'7a + 2w'ayS = 0,
be the same as that represented in areal co-ordinates by the equation
\3^ -\- yiy^ -h z// + ^Xyz + ^^Lzx -\- 2vxy = 0 ;
then, since — = ?^, ~ — , we have aa 0^ cy
U _ V _ w u' _ V __ w'
\(i^ fjbb^ v(? \'bc fi'ca vah'
Hence we can obtain the relation between the coefficients in the areal equation which corresponds to any given relation between the coefficients in the trilinear equation.
For example, the condition that uo? + v^"^ + w^f = 0 may be a rectangular hyperbola isw + v + w? = 0; hence the condition that \x^ + fiy'^ -{-vz^ = Q may be a rectangular hyperbola is \a^ + At6^ + vc^ = 0.
TANGENTIAL CO-ORDINATES.
293. If I, m, n be the three constants in the tri- linear or areal equation of any straight line, the position of the line will be determined when I, m and n are given ; and by changing the values of I, m, and n the equation may be made to represent any straight line whatever.
The quantities I, m, and n which thus define the position of a straight line are called the co-ordinates of the line.
If the equation of a straight line in areal co-ordinates be lx-\- my + nz=^0,
the lengths of the perpendiculars on the line from the angular points of the triangle of reference will be pro- portional to I, m, n. This follows at once from Art. 257; we will however give an independent proof.
Let the lengths of the perpendiculars from the angular points Ay By 0 of the triangle of reference be p, q, r
304 TANGENTIAL EQUATIONS.
respectively. Let the line cut BG in the point K, and let the co-ordinates of K be 0, y'y z.
Then q : r :: BK : OK ::-z : y.
But, since K is on the line, my' -^nz =0; therefore q : r :: m : n.
294. The lengths of the perpendiculars on a straight line from the angular points of the triangle of reference may be called the co-ordinates of the line. If any two of these perpendiculars be drawn in different directions they must be considered to have different signs.
From the preceding Article we see that the equation of a line whose co-ordinates are p, q, r is px + qy-\-rz= 0.
When the lengths of two of the perpendiculars on a straight line are given, there are two and only two positions of the line ; so that, when two of the co-ordinates of the line are given, the third has one of two particular values. Hence there must be some identical relation connecting the three co-ordinates of a line, and that relation must be of the second degree.
295. To find tJie identical relation which eodsts between the co-ordinates of any line.
Let 6 be the angle the line makes with BA, then we have q—p^csinO, and q — r — asin{d-\- B). The elimi- nation of 6 gives the required relation, viz.
a' {q -pf - lac cos B {q -p) (g - r) + c' {q - rf = 4A^ or «' {P -<l)ip- '^i + 6' (^ - r) {q-p)-\- c' (r -p) (r - ^) = 4A^
296. If the line px-{-qy + rz = 0 pass through a fixed point (/, g, h), then
p/+qg + rh = 0 (i).
So that the co-ordinates of all the lines which pass through the point whose areal co-ordinates are /, g, h satisfy the relation (i).
Hence the equation of a point is of the first degree.
TANGENTIAL EQUATIONS.
305
297. If the co-ordinates of a straight line are con- nected by any relation the line will envelope a curve, and the equation which expresses that relation is called the tangential equation of the curve.
We have seen that the tangential equation of a conic is of the second degree, and that every curve whose equation is of the second degree is a conic. If i/r (Z, m, %) = 0 be the tangential equation of the conic whose areal equation is </) {oc, y, z) = 0, and if the coefficients in the equation <^ = 0 be u, V, w, u\ V, w ; the corresponding coefficients in the equation 1/^ = 0 will be U, V, W, U', V\ W\ the minors of u, V, Wy u, v', w respectively in- the determinant
Uy W y
V
u , w
Since u, v, w, u, v\ w' are proportional to the minors of TJ, F, W, TJ\ V\ W in the determinant
jj , F', y
W, V, U' ; [See Art. 239]
F', U\ W
it follows that if yfr (I, m, n) = 0 be the tangential equa- tion of the conic whose areal equation is <^ (a;, y, z) = 0, then <f) {I, in, n) = 0 will be the tangential equation of the conic whose areal equation is yjr (a;, y, z)=0.
298. We can find the equation of the point of contact of any tangent by an investigation similar to that in Art. 261.
The equation is
Pdp'^^dq'^ dr "'
â– ^ dp ^ aq dr
'where <^ (p, q, r) is the equation of the conic, and p\ q, r are the co-ordinates of the tangent.
s. c. s. 20
306
EXAMPLES.
of the curve is
0.
If {p\ q, r) be not a tangent to the curve, the above equation will be the equation of the pole of (p', q, r).
The centre is the pole of the line at infinity whose co-ordinates are 1, 1, 1 ; hence the equation of the centre
dp dq dr
299. We shall conclude this chapter by the solution of some examples.
(1) If the sides of two triangles touch a given conic, their six angular points will lie on another conic.
Take one of the triangles for the triangle of reference. Let the equation of the given conic be
Let the equations of the sides of the second triangle be l-^a + mjP + n^y = 0, l^a. + m^ + n^y — Q, and Zsa + 7»3/3 + W37=0.
Then L {l^a + mjj/S + n^y) {1^0. + wig/S + W37) + M {IgCi, + m^ + n^y) {l^a + m^p + n^y)
+ N {l^a + WjjS + n^y) {l^a + m^^ + n.f^) = 0 will be the general equation of a conic circumscribing the triangle formed by these straight lines.
This conic will pass through the angular points of the triangle of reference if the coefficients of a^, /g^ and 7^ are all zero. That is, if L l^l^ + Ml^lj^ + Nljl.2=0, Lm^m^ + Mm^m-y + Nrn^rn^ = 0, and -£^2^3 + Mn.^nj^ + Nn-^n^ = 0.
Eliminating L, M, N, we see that the condition to be satisfied is
|
1,1, . |
^3^1 . |
hh |
|
m^m,, |
m,m^, |
riij^m.^ |
|
n^n, , |
W3?ll , |
WjTlj |
= 0, or
1^ Z2 1_
1
= 0.
But, since the three lines touch the given conic, we have
0,
■r+— +
= 0, and r + — +
= 0.
7^3
Eliminating X, /a, v, we see that the required condition is satisfied.
[See also Art. 322, Ex. 2.]
EXAMPLES. 307
(2) If one triangle can he inscribed in one conic loith its sides touching anotJier conic, then an infinite number of triangles can be so described.
Let ABC be the triangle whose angular points are on the conic S, and whose sides touch the conic S.
Let any other tangent to S be drawn cutting S in the points B', C\ and let the other tangents to S through B', C meet at A'. Then A'B'C and ABC are two triangles whose sides touch S. Therefore by the pre- ceding question the six points A, B, 0, B', C, A' are on a conic. But five of the points, viz. -4, J5, C, B', C, are on the conic S, and only one conic will pass through five points, therefore C also is on S.
(3) Four circles are described so that each of the four triangles, formed by each three of four given straight lines, is self-polar with respect to one of the circles; prove that these four circles and the circle circumscribing the triangle formed by the diagonals of tJie quadrilateral have a common radical axis.
Take the triangle formed by the diagonals for the triangle of reference, then the equations of the four straight lines will be Za±m/3±W7 = 0. All conies with respect to which the lines
Za + mj8 + %7 = 0, la-m^-{-ny = 0, and la-\-m^-ny = 0 form a self-polar triangle are included in the equation
L{la + m^ + nyf + M {la-mp + nyf + N {la + m^-nyf^O (i).
If this conic be a circle its equation can be put in the form
a^y + &7a + c a/3 + (\a + /A/S + ry ) ( a a + ftjS -t- C7) = 0 (ii) ,
and Xa-F/ij3 + »'7 = 0 is the radical axis of (ii) and of the circumscribing circle. Comparing coefficients of a^, jS^ ^jj^ ^a Jq ^jj ^nd (ii) we obtain, for the equation of the radical axis
This is clearly the same for all four circles.
(4) A line cuts two given conies in P, P', and Q, Q', so that the range â– P> Q. -P'j Q' is harmonic; find the envelope of the line.
Refer the conies to their common self-conjugate triangle and let their equations be
ua^ + v^^ + wy^=0, u'a^ + v'p^ + iD'y^=0. Let the equation of the line be
la + m^ + 7iy=0. Then the lines AP, AP" are given by the equation u (m/3 + nyY + Ih)^^ + Ihoy^ = 0, or ^ {urn^ + vP) + 2umnpy + {un^ + wl^) 7^ = 0.
20—2
308 EXAMPLES.
And similarly ^Q, ^^ are given by
j82 (w'm2 + v'P) + 2u'mnpy + [u'ri^ + w'V^) 7^ = 0. If therefore A {PQP'Q'] is harmonic, we must have [Art. 58]
(w?;i2 + vP) (wV + w'V) + (un^ + wP) {u'vP + v'P) = 2uu'm^iP ; which reduces to
(vw' + wv') P + {wu' + uio') rrP + {uv' + vu') n^ = 0. This condition shews that the line always touches the conic «2 02 y
vw' + wi; t(;w + WW? MV + vu It is easy to shew that the envelope touches the eight tangents to the given conies at their four points of intersection.
(5) The director-circles of all conies which are inscribed in the same quadrilateral Jiave a common radical axis.
Let the triangle formed by the diagonals of the quadrilateral be taken for the triangle of reference.
Then the equations of the four lines will be la±mp±ny=0. [Art. 259.]
The equation of any one of the conies will be tia^ + v^ + wy^=0. [Art. 281.]
The equation of the two tangents from the point (a'^y') is
(Ma2 + v^^ + wy^) {ua'^ + v^"^ + loy'^) - (ua'a + r/3'y3 + wy'y)^ = 0.
The condition that these Unes may be perpendicular is [Art. 268] It (Vj8'2 + 'wy'^) + V (W7'2 + ita'2) + w (wa'2 + vp) + 2vw ^'y' cos A
+ Iwuy'fx! cos B + 2uva'^' cos C = 0 .
Hence the equation of the director-circle of the conic ua? + v^^ + wy^ = 0 will be /3^ + 7^ + 2)87 cos ^ 72 + a2 4.27acos.g a^ + ^+2a^Q0BC _
u V w ~ ""^
But, since the conic touches the four lines Za±wi/S± 717=0, we have
P m2 w2
-+— + -=0 (ii).
Comparing (i) and (ii) we see that all the director-circles pass through the points given by
^32 + 72 + 2/37COS A ^y"- + a^-\-2ya,G0BB _a?+§^ + 2a/Seos G P m^ ~ n^ '
[See also Art. 243, Ex. (2), and Art. 307.]
(6) To find the tangential equation of the circle loith respect to tchich the triangle of reference is self -polar.
The trilinear equation of the circle is
a^a COS A + /326 cos B + 72c cos C = 0.
EXAMPLES ON CHAPTER XIII. 309
The line la + in^ + ny=0 will touch the circle, if
+ r— r^+ ^=0.
a cos A h cos B c cos (7 If p, q, r be the perpendiculars on the line from the angular points of the triangle
^ = l=-rArt. 2571. I VI n
a h c
Hence from the condition of tangency
p^ tan A + 2"^ tan B + r^ tan (7= 0,
which is the required tangential equation.
Examples on Chapter XIII.
1. Shew that the minor axis of an ellipse inscribed in a given triangle cannot exceed the diameter of the inscribed circle.
2. Find the area of a triangle in terms of the trilinear or areal co-ordinates of its angular points.
3. If four conies have a common self-conjugate triangle, the four points of intersection of any two and the four points of intersection of the other two lie on a conic.
4. Shew that the eight points of contact of two conies with their common tangents lie on a conic.
5. Shew that the eight tangents to two conies at their common points touch a conic.
6. Any three pairs of points which divide the three diagonals of a quadrilateral harmonically are on a conic.
7. Find the equation of the nine-point circle by considering it as the circle circumscribing the triangle formed by the lines
aa — h^ — cy~ 0, b^-cy — aa = 0, and cy — aa — b^^ 0.
8. Shew that the equation of the circle concentric with afty + bya + ca/3 = 0 and of radius r is
a^y + bya + cap j — {aa + 6)8 + CyY = 0,
where i? is the radius of the circle circumscribing the triangle of reference.
310 EXAMPLES ON CHAPTER XIII.
9. The equation of the circumscribing conic, whose diameters parallel to the sides of the triangle of reference are
ah c -
10. ABC is a triangle inscribed in a conic, and the tangents to the conic at ^, j5, C are B'G\ C'A\ A'B' respectively ; shew that AA\ BB\ and CC meet in a point. Shew also that, if B be the point of intersection of BG^ B'G' ; E the point of inter- section of GA^ C'A\ and F the point of intersection of AB, AB'-j Z), E, F will be a straight line.
11. Lines are drawn from the angular points A, B^ G oi a. triangle through a point P to meet the opposite sides in A', B', C'. B'G' meets BG in K, G'A' meets GA in Z, and A'B' meets AB in M. Shew that ZT, Z, M are on a straight line. Shew also (i) that if P moves on a fixed straight line then KLM will touch a conic inscribed in 'the triangle ABC ; (ii) that if P moves on a fixed conic circumscribing the triangle ABG, then KLM will pass through a fixed point ; (iii) that if P moves on a fixed conic touching two sides of the triangle where they are met by the third, KLM will envelope a conic.
12. Lines drawn through the angular points A, B, G of & triangle and through a point 0 meet the opposite sides in A'f B\ C"; and those drawn through a point 0' meet the 0])posite sides in A", B", G". If P be the point of intersection oiB'C' and B"G'\^ Q be the point of intersection of G'A', G"A", and R be the point of intersection of A'B', A"B" ; shew that AP, BQf GR will meet in some point Z. Shew also that, if 0, 0' be any two points on a fixed conic through A, B, G, the point Z will be fixed.
13. The locus of the pole of a given straight line with respect to a system of conies through four given points is a conic which passes through the diagonal-points of the quad- rangle formed by the given points.
14. The envelope of the polar of a given point with respect to a system of conies touching four given straight lines is a conic which touches the diagonals of the quadrilateral formed by the given lines.
EXAMPLES ON CHAPTER XIII. 311
15. Shew that the locus of the points of contact of tangents, drawn parallel to a fixed line, to the conies in- scribed in a given quadrilateral, is a cubic; and notice any remarkable points, connected with the quadrilateral, through which the cubic passes.
16. An ellipse is inscribed within a triangle and has its centre at the centre of the circumscribing circle. Shew that its major and minor axes are R + d and R — d respectively, R being the radius of the circumscribing circle and d the distance between the centre and the orthocentre.
17. Prove that a conic circumscribing a triangle ABC will be an ellipse if the centre lie within the triangle DEF or within the angles vertically opposite to the angles of the triangle DEF, where i>, E, F are the middle points of the sides of the triangle ABC.
18. Shew that the locus of the foci of parabolas to which the triangle of reference is self -polar is the nine-point circle.
19. Shew that the locus of the foci of all conies touching the four lines la, ± m/3 ± ny = 0 is the cubic
pa pa p2 p2
ij + 2 + z^ + £i - 0
la + TTh^ + ny la — mp — ny —la+m^—ny -la—m^-\-ny ' where P^^ = l^ + m^ + n^- 2mn cos A - 2nl cos B -2lm cos C, and Pg*, jPg", F^^ have similar values.
20. If a conic be inscribed in a given triangle, and its major axis pass through the fixed point (yj ^, /«.), the locus of its focus is the cubic
/a {^ - /) + ffl3 (/ - a') + hy (a' - y3») = 0.
21. If the centre of a conic inscribed in a triangle move along a fixed straight line, the foci will lie on a cubic circum- scribing the triangle.
22. The locus of the centres of the rectangular hyperbolas with respect to which the triangle of reference is self-conjugate is the circumscribing circle.
23. The locus of the centres of all rectangular hyperbolas inscribed in the triangle of reference is the self -conjugate circle.
812 EXAMPLES ON CHAPTER XIII.
24. Shew that the nine-point circle of a triangle touches the inscribed circle and each of the escribed circles.
25. The tangents to the nine-point circle at the points where it touches the inscribed and escribed circles form a quadrilateral, each diagonal of which passes through an angular point of the triangle, and the lines joining corresponding angular points of the original triangle and of the triangle formed by the diagonals are all parallel to the radical axis of the nine-point circle and the circumscribing circle.
26. The polars of the points A, B, C with respect to a conic are B'C\ G'A\ A'B' respectively; shew that AA\ BB\ CC meet in a point.
27. If an equilateral hyperbola pass through the middle points of the sides of a triangle ABC and cuts the sides BC, OA, AB again in a, )8, y respectively, then Aa, Bp, Cy meet in a point on the circumscribed circle of the triangle ABC.
28. Shew that the locus of the intersection of the polars of all points in a given straight line with respect to two given conies is a conic circumscribing their common self-conjugate triangle.
29. Two conies have double contact ; shew that the locus of the poles with respect to one conic of the tangents to the other is a conic which has double contact with both at their common points.
30. Two triangles are inscribed in a conic ; shew that their six sides touch another conic.
31. Two triangles are self-polar with respect to a conic ; shew that their six angular points are on a second conic, and that their six sides touch a third conic.
32. If one triangle can be described self -polar to a given conic and with its angular points on another given conic, an infinite number of triangles can be so described.
33. A system of similar conies have a common self-conju- gate triangle ; shew that their centres are on a curve of the 4th degree which passes through the circular points at infinity and of which the angular points of the triangle are double points.
EXAMPLES ON CHAPTER XIII. 313
34. If A, B, C, A', B', a be six points such that AA\ BB\ CC meet in a point, then will the six straight lines AB\ AC, BC\ BA\ CA' and CB' touch a conic.
35. A conic is inscribed in a triangle and is such that the normals at the points of contact meet in a point; prove that the point of concurrence describes a cubic curve whose asymptotes are perpendicular to the sides of the triangle.
36. If p^, ^g, p^, p^ be the lengths of the perpendiculars drawn from the vertices A, B, C, J) oi a. quadrilateral circum- scribed about a conic on any other tangent to the conic, shew that the ratio oip^p^ ^^ P2P4: ^^^^ ^® constant.
37. The polars with respect to any conic of the angular points A, B, C of a triangle meet the opposite sides in A\ B\ C ; shew that the circles on ^^', BB\ CC as diameters have a common radical axis.
38. A parabola touches one side of a triangle in its middle point, and the other two sides produced ; prove that the per- pendiculars drawn from the angular points of the triangle upon any tangent to the parabola are in harmonical pro- gression.
39. Shew that the tangential equation of the circum- scribing circle is a Jp + b Jq + c Jr = 0. Hence shew that the tangential equation of the nine-point circle is
aj(q + r) + b J{r +p) + c J{p + q).
40. The locus of the centre of a conic inscribed in a given triangle, and having the sum of the squares of its axis constant, is a circle.
41. The director circles of all conies inscribed in the same triangle are cut orthogonally by the circle to which the triangle of reference is self-polar.
42. The circles described on the diagonals of a complete quadrilateral are cut orthogonally by the circle round the triangle formed by the diagonals.
43. If three conies circumscribe the same quadrilateral, shew that a common tangent to any two is cut harmonically by the third.
314 EXAMPLES ON CHAPTER XIII.
44. If three conies are inscribed in the same quadrilateral the tangents to two of them at a common point and the tan- gents to the third from that point form a harmonic pencil.
45. The locus of a point the pairs of tangents from which to two given conies form a harmonic pencil is a conic on which lie the eight points in which the given conies touch their com- mon tangents.
46. The locus of a point from which the tangents drawn to two equal circles form a harmonic pencil is a conic, which is an ellipse if the circles cut at an angle less than a right angle, and two parallel sti-aight lines if they cut at right angles.
47. A triangle is circumscribed about one conic and two of its angular points are on a second conic; find the locus of the third angular point.
48. A triangle is inscribed in one conic and two of its sides touch a second conic ; find the envelope of the third side.
49. The angular points of a triangle are on the sides of a given triangle, and two of its sides pass through fixed points ; shew that the third side will envelope a conic.
50. From the angular points of the fundamental tiiangle pairs of tangents are drawn to (uvwu'v'w'^xyzY = 0^ and each pair determine with the opposite sides a pair of points. Find the equation to the conic on which these six points lie, and shew that the conic
Jx (v'w' — uu') + Jy {w'v! - vv') + ^% {u'v - ww') = 0 and the above two conies have a common inscribed quadri- lateral.
CHAPTER XIV. Reciprocal Polars. Projections.
800. If we have any figure consisting of any number of points and straight lines in a plane, and we take the polars of those points and the poles of the lines, with respect to a fixed conic (7, we obtain another figure which is called the polar reciprocal of the former with respect to the auxiliary conic C.
When a point in one figure and a line in the reciprocal figure are pole and polar with respect to the auxiliary conic G, we shall say that they correspond to one another. If in one figure we have a curve 8 the lines which corre- spond to the different points of 8 will all touch some curve S'. Let the lines corresponding to the two points P, Q oi 8 meet in T ; then T is the pole of the line PQ with respect to G, that is the line PQ corresponds to the point T. Now, if the point Q move up to and ultimately coincide with P, the two corresponding tangents to 8' will also ultimately coincide with one another, and their point of intersection T will idtimately be on the curve >S^'. So that a tangent to the curve 8 corresponds to a point on the curve 8\ just as a tangent to >Si' corresponds to a point on >S^. Hence we see that 8 is generated from 8' exactly as 8' is from 8.
301. If any line L cut the curve 8 in any number of points P, Q, P...we shall have tangents to >S" corresponding to the ^om^5 P, Q, P..., and these tangents will all pass through a point, viz. through the pole of L with respect to the auxiliary conic. Hence as many tangents to >Si' can be drawn through a point as there are points on 8 lying on a
316 RECIPROCAL THEOREMS.
straight line. That is to say the class [Art. 240] of S' is equal to the degree of S. Reciprocally the degree of B' is equal to the class of B.
In particular, if >S^ be a conic it is of the second degree, and of the second class. Hence the reciprocal curve is of the second class, and of the second degree, and is therefore also a conic.
302. To find the polar reciprocal of one conic with respect to another.
Let the equation of the auxiliary conic be
_ ax^ + Py' + l=0 ..(i);
and let the equation of the conic whose reciprocal is required be
ax" + 63/' + c + 2/y + 2gx + 2hxy = 0 . . . (ii) . The line Ix + my + n — O will touch (ii) if AV + Bm^ + Cn- + "IFmn + ^Onl + ^Hlm = 0 (iii). And, if the pole of Ix + my + n = i) with respect to (i) be {x\ y), its equation is the same as aafx + ^y'y + 1=0.
Therefore — -, = tt— , = - .
ax py 1
Substitute, in (iii), and we have
AaV + B^Y + C' + ^F^y' + ^Goix' + ^Ha^x'y' = 0.
Hence the locus of the poles with respect to (i) of
tangents to (ii) is the conic whose equation is
^aV + B^Y + C' + ^F^y + max + 2Ha^xy = 0.
303. The method of Reciprocal Polars enables us to obtain from any given theorem concerning the positions of points and lines, another theorem in which straight lines take the place of points and points of straight lines. Before proceeding to give examples of such reciprocal theorems we will give some simple cases of correspondence.
Points in one figure correspond to straight lines in the reciprocal figure.
The line joining two points in one figure corresponds to the point of intersection of the corresponding lines in the other.
RECIPROCAL THEOREMS. 317
The tangent to any curve in one figure corresponds to a point on the corresponding curve in the reciprocal figure.
The point of contact of a tangent corresponds to the tangent at the corresponding point.
If two curves touch, that is have two coincident points common, the reciprocal curves will have two coincident tangents common, and will therefore also touch.
The chord joining two points corresponds to the point of intersection of the corresponding tangents.
The chord of contact of two tangents corresponds to the point of intersection of tangents at the corresponding points.
Since the pole of any line through the centre of the auxiliary conic is at infinity, we see that the points at infinity on the reciprocal curve correspond to the tangents to the original curve from the centre of the auxiliary conic. Hence the reciprocal of a conic is an hyperbola, parabola, or ellipse^ according as the tangents to it from the centre of the auxiliary conic are real, coincident, or imaginary; that is according as the centre of the auxiliary conic is outside, upon, or within the curve.
The following are examples of reciprocal theorems.
If the angular points of two If the sides of two triangles
triangles are on a conic, their six touch a conic, their six angular
sides will touch another conic. points are on another conic.
The three intersections of oppo- The three lines joining opposite
site sides of a hexagon inscribed in angular points of a hexagon de-
a conic lie on a straight line. scribed about a conic meet in a
{PascaVs Theorem). point. {Brianchon's Theorem).
If the three sides of a triangle If the three angular points of a
touch a conic, and two of its angu- triangle lie on a conic, and two of
lar points lie on a second conic, the its sides touch a second conic, the
locus of the third angular point is envelope of the third side is a
a conic. conic.
If the sides of a triangle touch If the angular points of a tri-
a conic, the three lines joining an angle lie on a conic, the three points
angular point to the point of con- of intersection of a side and the
tact of the opposite side meet in a tangent at the opposite angular
point. point lie on a line.
318 RECIPROCATION WITH RESPECT TO A CIRCLE.
The polars of a given point with The poles of a given straight
respect to a system of conies through Hne with respect to a system of
four given points all pass through a eouics touching four given straight
fixed point. lines all lie on a fixed straight line.
The locus of the pole of a given The envelope of the polar of a
line with respect to a system of given point with respect to a system
conies through four fixed points is a of conies touching four fixed lines
conic. is a conic.
304. We now proceed to consider the results which can be obtained by reciprocating with respect to a circle.
We know that the line joining the centre of a circle to any point P is perpendicular to the polar of P with respect to the circle. Hence, if P, Q be any two points, the angle between the polars of these points with respect to a circle is equal to the angle that PQ subtends at the centre of the circle. Reciprocally the angle between any two straight lines is equal to the angle which the line joining their poles with respect to a circle subtends at the centre of the circle.
We know also that the distances, from the centre of a circle, of any point and of its polar with respect to that circle, are inversely proportional to one another.
If we reciprocate with respect to a circle it is clear that a change in the radius of the auxiliary circle will make no change in the shape of the reciprocal curve, but only in its size. Hence, if we are not concerned with the absolute magnitudes of the lines in the reciprocal figure, we only require to know the centre of the auxiliary circle. We may therefore speak of reciprocating with respect to a point 0, instead of with respect to a circle having 0 for centre.
305. If any conic be reciprocated with respect to a point 0, the points on the reciprocal curve which corre- spond to the tangents through 0 to the original curve must be at an infinite distance.
The directions of the lines to the points at infinity on the reciprocal curve are perpendicular to the tangents from 0 to the original curve ; and hence the angle between
RECIPROCATION WITH RESPECT TO A CIRCLE. 319
the asymptotes of the reciprocal curve is supplementary to the angle between the tangents from 0 to the original curve.
In particular, if the tangents from 0 to the original curve be at right angles, the reciprocal conic will be a rectangular hyperbola.
The axes of the reciprocal conic bisect the angles between its asymptotes. The axes are therefore parallel to the bisectors of the angles between the tangents from 0 to the original conic.
Corresponding to the points at infinity on the original conic we have the tangents to the reciprocal conic which pass through the origin. Hence the tangents from the origin to the reciprocal conic are perpendicular to the directions of the lines to the points at infinity on the original conic, so that the angle between the asymptotes of the original conic is supplementary to the angle between the tangents from the origin to the reciprocal conic.
In particular, if a rectangular hyperbola 'be recipro- cated with respect to any point 0, the tangents from 0 to the reciprocal conic will be at right angles to one another; in other words 0 is a point on the director-circle of the reciprocal conic.
806. The reciprocal of the origin is the line at infinity, and therefore the reciprocal of the polar of the origin is the pole of the line at infinity. That is to say, the polar of the origin reciprocates into the centre of the reciprocal conic.
307. Ab an example of reciprocation take the known theorem — "If two of the conies which pass through four given points are rectangular hyperbolas, they will all be rectangular hyperbolas." If this be reciprocated with respect to any point 0 we obtain the following, " If the director-circles of two of the conies which touch four given straight lines pass through a point 0, the director-circles of all the conies will pass through 0." Whence we have " The director-circles of all conies which touch four given straight lines have a given radical axis."
320 RECIPROCAL OF A CIRCLE.
308. To find the polar reciprocal of one circle with respect to another.
Let G be the centre and a be the radius of the circle to be reciprocated, 0 the centre and k the radius of the auxiliary circle, and let c be the distance between the centres of the two circles.
Let PiV.be any tangent to the circle C, and let P' be its pole with respect to the auxiliary circle. Let OP' meet the tangent in the point K, and draw GM perpen- dicular to OiV.
Then OF.ON=]^',
... ^, = ON^OM + MN = cco^GOM + a. Hence the equation of the locus of P' is
_= 1 +- cos^. r a
This is the equation of a conic having 0 for focus, — for semi'latus rectum, and - for eccentmcity. The direc- trix of the conic is the line whose equation is
A; yj Ic
~ =c cos u. or a; = — . r c
Hence the directrix of the reciprocal curve is the polar of the centre of the original circle.
RECIPROCAL OF A. CIRCLE. 321
It is clear from the value found above for the eccen- tricity, that the reciprocal curve is an ellipse if the point 0 be within the circle G, an hyperbola if 0 be outside that circle, and a parabola if 0 be upon the circumference of the circle.
Ex. 1. Tangents to a conic subtend equal angles at a focus.
Reciprocate with respect to the focus: — then corresponding to the two tangents to the conic, there are two points on a circle ; the point of intersection of the tangents to the conic corresponds to the line joining the two points on the circle ; and the points of contact of the tangents to the conic correspond to the tangents at the points on the circle. Also the angle subtended at the focus of the conic by any two points is equal to the angle between the hues corresponding to those two points. Hence the reciprocal theorem is — The line joining two points on a circle makes equal angles with the tangents at those points.
Ex. 2. The envelope of the chord of a conic lohich subtends a right angle at a fixed point 0 is a conic having 0 for a focus, and the polar of 0, with respect to the original conic, for the corresponding directrix.
Reciprocate with respect to O, and the proposition becomes— The locus of the point of intersection of tangents to a conic which are at right angles to one another is a concentric circle.
Ex. 3. If two conies have a common focus, two of their common chords will pass through the intersection of their directrices.
Reciprocate with respect to the common focus, and the proposition becomes — Two of the points of intersection of the common tangents to two circles are on the line joining the centres of the circles.
Ex. 4. The ortlwcentre of a triangle circumscribing a parabola is on the directrix.
Reciprocating with respect to the orthocentre we obtain — A conic circumscribing a triangle and passing through the orthocentre is a rect- angular hyperbola.
Many of the examples on Chapter VIIL are easily proved by reciproca- tion : for example, the reciprocal of 23 with respect to the common focus is — circles are described with equal radii, and with their centres on a second circle ; prove that they all touch two fixed circles, whose radii are the sum and difference respectively of the radii of the moving circle and of the second circle, and which are concentric with the second circle.
S. C. S. 21
S22 KECIPROCAL OF CONFOCAL CONICS.
309. If we have a system of circles with the same radical axis we can reciprocate them into a system of confocal conies.
If we reciprocate with respect to any point 0 we obtain a system of conies having 0 for one focus, and [Art. 306] the centre of any conic is the reciprocal of the polar of 0 with respect to the corresponding circle. Now either of the two 'limiting points' of the system is such that its polar with respect to any circle of the system is a fixed straight line, namely a line through the other limiting point parallel to the radical axis. If therefore the system of circles be reciprocated with respect to a limiting point the reciprocals will have the same centre ; and if they have a common centre and one common focus they will be confocal. Since the radical axis is parallel to and midway between a limiting point and its polar, the re- ciprocal of the radical axis (with respect to the limiting point) is on the line through the focus and centre of the reciprocal conies, and is twice as far from the focus as the centre ; so that when we reciprocate a system of coaxial circles with respect to a limiting point, the radical axis reciprocates into the other focus of the system of confocal conies.
The following theorems are reciprocal :
The tangents at a common The points of contact of a corn-
point of two confocal conies are at mon tangent to two circles subtend right angles. a right angle at one of the limit-
ing points.
The locus of the point of inter- The envelope of the line joining
section of two lines, each of which two points, each of which is on one touches one of two confocal conies, of two circles, and which subtend and which are at right angles to a right angle at a limiting point, one another, is a circle. is a conic one of whose foci is at
the limiting point.
If from any point two pairs of If any straight line cut two
tangents P, P' and Q, Q' be drawn circles in the points P, P' and to two confocal conies; the angle Q, Q' \ the angles subtended at a between P and Q is equal to that limiting point by PQ and P'Q' are between P' and Q'. equal.
PROJECTION. 323
From any point four tangents Any line cuts two circles in P,
P, i*' and Q, Q' are drawn to two P' and Q, Q' respectively ; and the
confocal conies, and the point of tangent at P cuts the tangents at
contact of P is joined to the points Q, Q' in q, q'; shew that Pq, Pq'
of contact of Q, Q' ; shew that these subtend equal (or supplementary)
lines make equal angles with the angles at a limiting point, tangent P. [Art. 229.]
Projection.
310. If any point P be joined to a fixed point F, and VP be cut by any fixed plane in P\ the point P' is called the projection of P on that plane. The point V is called the vertex or the centre of projection, and the cutting plane is called the plane of projection.
311. The projection of any straight line is a straight line.
For the straight lines joining V to all the points of any straight line are in a plane, and this is cut by the plane of projection in a straight line.
312. Any plane curve is projected into a curve of the same degree.
For, if any straight line meet the original curve in any number of points A, B, C, D..., the projection of the line will meet the projection of the curve where VA, VB, VG, VD... meet the plane of projection. There will therefore be the same number of points on a straight line in the one curve as in the other. This proves the proposition.
In particular, the projection of a conic is a conic.
This proposition includes the geometrical theorem that every plane section of a right circular cone is a conic.
313. A tangent to a curve projects into a tangent to the projected curve.
For, if a straight line meet a curve in two points A, B, the projection of that line will meet the projected curve in two points a, h where VA, VB meet the plane of pro- jection. Now if A and B coincide, so also will a and b.
21—2
324 PROJECTION.
314. The relation of pole and polar with respect to a conic are unaltered hy projection.
This follows from the two preceding Articles.
It is also clear that two conjugate points, or two con- jugate lines, with respect to a conic, project into conjugate points, or lines, with respect to the projected conic.
315. Draw through the vertex a plane parallel to the plane of projection, and let it cut the original plane in the line K'L'. Then, since the plane VK'L' and the plane of projection are parallel, their line of intersection, which is the projection of K'L\ is at an infinite distance.
Hence to project any particular straight line K'L' to an infinite distance, take any point V for vertex and a plane parallel to the plane VK'L' for the plane of pro- jection.
Straight lines which meet in any point on the line K'L' will be projected into parallel straight lines, for their point of intersection will be projected to infinity.
316. A system of parallel lines on the original plane will be projected into lines which meet in a point.
For, let VP be the line through the vertex parallel to the system, P being on the plane of projection; then, since VP is in the plane through V and any one of the parallel lines, the projection of every one of the parallel lines will pass through P.
For different systems of parallel lines the point P will change; but, since VP is always parallel to the original plane, the point P is always on the line of intersection of the plane of projection and a plane through the vertex parallel to the original plane.
Hence any system of parallel lines on the original plane is projected into a system of lines passing through a point, and all such points, for different systems of parallel lines, are on a straight line.
317. Let KL be the line of intersection of the original plane and the plane of projection. Draw through the vertex a plane parallel to the plane of projection, and let
PROJECTION. 325
it cut the original plane in the line K'L' . Let the two straight lines AOA\ BOB' meet the lines KL, K'L' in the points A, B and A', B' respectively; and let VO meet the plane of projection in 0' . Then AO' and BO' are the projections oi AOA' and BOB.
Since the planes VA'B', AO'B are parallel, and parallel planes are cut by the same plane in parallel lines, the lines VA', VB' are parallel respectively to AO', BO', The angle A' VB' is therefore equal to the angle AO'B, that is, A' VB' is equal to the angle into which AOB is projected.
Similarly, if the straight lines CD, ED, meet K'L' in C, D' respectively, the angle C VB' will be equal to the angle into which CDE is projected.
From the above we obtain the fundamental proposition in the theory of projections, viz.,
Any straight line can he projected to infinity, and at the same time any two angles into given angles.
For, let the straight lines bounding the two angles meet the line which is to be projected to infinity in the points
326 PROJECTION.
A', B' and C\ D' \ draw any plane through A'B'G'I)\ and in that plane draw segments of circles through A\ B' and C", D' respectively containing angles equal to the two given angles. Either of the points of intersection of these segments of circles may be taken for the centre of pro- jection, and the plane of projection must be taken parallel to the plane we have drawn through A'B'G'U.
If the segments do not meet, the centre of projection is imaginary.
Ex. 1. To shew that any quadrilateral can he projected into a square.
Let ABGD be the quadrilateral j and let P, Q [see figure to Art. 60] be the points of intersection of a pair of opposite sides, and let the diago- nals BD, AC meet the line PQ in the points S, R. Then, if we project PQ to infinity and at the same time the angles PDQ and ROS into right angles, the projection must be a square. For, since PQ is projected to infinity, the pairs of opposite sides of the projection will be parallel, that is to say, the projection is a parallelogram ; also one of the angles of the parallelogram is a right angle, and the angle between the diagonals is a right angle ; hence the projection is a square.
Ex. 2. To shew that the triangle formed by the diagonals of a quad- rilateral is self-polar with respect to any conic which touches the sides of the quadrilateral*
Project the quadrilateral into a square; then, the circle circumscribing the square is the director-circle of the conic, therefore the intersection of the diagonals of the square is the centre of the conic.
Now the polar of the centre is the line at infinity ; hence the polar of the point of intersection of two of the diagonals is the third diagonal.
Ex. 3. If a conic he inscribed in a quadrilateral the line joining tioo of the points of contact will pass through one of the angular points of the triangle formed by the diagonals of the quadrilateral.
Ex. 4. If ABC be a triangle circumscribing a parabola, and the parallelograms ABA'C, BOB' A, and CAC'B be completed ; then the chords of contact will pass respectively through A', B\ C".
This is a particular case of Ex. 3, one side of the quadrilateral being the line at infinity.
Ex. 5. If the three lines joining the angular points of two triangles meet in a point, the three points of intersection of coiresponding sides will lie on a straight line.
ANY CONIC PROJECTED INTO A CIRCLE. 327
Project two of the points of intersection of corresponding sides to infinity, then two pairs of corresponding sides will be parallel, and it is easy to shew that the third pair will also be parallel.
Ex. 6. Any tico conies can he projected into concentric conies. [See Art. 283.]
318. A717/ conic can he j^f^ojected into a circle having the projection of any given point for centre.
Let 0 be the point whose projection is to be the centre of the projected curve.
Let P be any point on the polar of 0, and let OQ be the polar of P; then OP and OQ are conjugate lines.
Take OP', OQ' another pair of conjugate lines.
Then project the polar of 0 to infinity, and the angles POQ, P'OQ' into right angles. We shall then have a conic whose centre is the projection of 0, and since two pairs of conjugate diameters are at right angles, the conic is a circle.
319. A system of conies inscribed in a quadrilateral can he projected into confocal conies.
Let two of the sides of the quadrilateral intersect in the point A, and the other two in the point B. Draw any conic through the points A, B, and project this conic into a circle, the line AB being projected to infinity; then, A, B are projected into the circular points at infinity, and since the tangents from the circular points at infinity to
328 EXAMPLES.
all the conies of the system are the same, the conies must be confocal.
Ex. 1. Conies through four given points can "be projected into coaxial circles.
For, project the line joining two of the points to infinity, and one of the conies into a circle ; then all the conies will he projected into circles, for they all go through the circular points at infinity.
Ex. 2. Conies lohich have double contact with one another can he projected into concentric circles.
Ex. 3. The three points of intersection of opposite sides of a hexagon inscribed in a conic lie on a straight line. [Pascal's Theorem.]
Project the conic into a circle, and the line joining the points of inter- section of two pairs of opposite sides to infinity ; then we have to prove that if two pairs of opposite sides of a hexagon inscribed in a circle are parallel, the third pair are also parallel.
Ex. 4. Sheio that all conies through four fixed points can be pro- jected into rectangular hyperbolas.
There are three pairs of lines through the four points, and if two of the angles between these pairs of lines be projected into right angles, all the conies will be projected into rectangular hyperbolas. [Art. 187, Ex. 1.]
Ex. 5. Any three chords of a conic can be projected into equal chords of a circle.
Let AA\ BB', CC be the chords ; let AB\ A'B meet in K, and AC, A'C in L. Project the conic into a circle, KL being projected to infinity.
Ex. 6. If two triangles are self polar with respect to a conic, their six angular points are on a conic, and their six sides touch a conic.
Let the triangles be ABC, A'B'C. Project BC to infinity, and the conic into a circle ; then A is projected into the centre of the circle, and AB, AC are at right angles, since ABC is self polar; also, since A'B'C is self polar with respect to the circle, A is the orthocentre of the triangle A'B'C.
Now a rectangular hyperbola through A', B', C will pass through A, and a rectangular hyperbola through B will go through C. Hence, since a rectangular hyperbola can be drawn through any four points, the six points A, B, C, A', B', C are on a conic.
Also a parabola can be drawn to touch the four straight lines B'C, CA', A'B', AB. And A is on the directrix of the parabola [Art. 107 (3)] ; therefore -4 C is a tangent. Hence a conic touches the six sides of the two triangles.
CROSS RATIOS UNALTERED BY PROJECTION. 329
820. Properties of a figure which are true for any pro- jection of that figure are called projective properties. In general such properties do not involve magnitudes. There are however some projective properties in which the mag- nitudes of lines and angles are involved : the most impor- tant of these is the following : —
T7ie cross ratios of pencils and ranges are unaltered hy projection.
Let A, B, Gy D be four points in a straight line, and A\ B', G\ D' be their projections. Then, if V be the centre of projection, YAA', VBB\ VCG', YDD' are straight lines; and we have [Art. 55]
[ABGD] = V[ABGD] = [A'FG'D']. . If we have any pencil of four straight lines meeting in Oy and these be cut by any transversal in ^, J5, (7, D ; then 0 [ABGD] = [ABGD] = V{ABCD] = [AB'CU]
= 0' {A'FG'D'].
From the above together with Article 62 it follows that if any number of points he in involution, their projections will he in involution.
Ex. 1. Any cJiord of a conic through a given point 0 is divided harmonically hy the curve and the polar of 0.
Project the polar of 0 to infinity, then 0 is the centre of the projec- tion, the chord therefore is bisected in 0, and {POQoo } is harmonic when PO=OQ.
Ex. 2. Conies through four fixed points are cut hy any straight line in pairs of points in involution. [Desargue's Theorem].
Project two of the points into the circular points at infinity, then the conies are projected into co-axial circles, and the proposition is obvious.
321. The cross ratio of the pencil formed hy four intersecting straight lines is equal to that of the range formed hy their poles with respect to any conic.
Since the cross ratios of pencils and ranges are unaltered by projection, we may project the conic into a circle. Now in a circle any straight line is perpendicular to the line joining the centre of the circle to its pole with
330 ANHARMONIC PROPERTIES OF CONICS.
respect to the circle. Hence the cross ratio of the pencil formed by four intersecting straight lines is equal to that of the pencil subtended at the centre of the circle by their poles, and therefore equal to the cross ratio of the range formed by their poles.
322. Tlie cross ratio of the pencil formed hy joining any point on a conic to four fixed points is constant, and is equal to that of the range in which the tangents at those points are cut hy any tangent.
Since the cross ratios of pencils and ranges are un- altered by projection, we need only prove the proposition for a circle.
Let A, B, C, Dhe four fixed points on a circle ; let P be any other point on the circle, and let the tangent at P meet the tangents Sit A, B,C,D in the points A\ B\ C, D'.
Then, if 0 be the centre of the circle, OA! is perpen- dicular to PA, OB' to PB, OO to PC, and OD' to PD.
Hence
[A'ECU] = 0 [A'B'C'D']=P[ABGD].
But the angles APB, BPC, CPD are constant, since A, B, G, D are fixed points. Therefore {A'B' CD'] = P [ABGD] = const.
If Q be any point which is not on the circle, Q [ABGD] cannot be equal to P {ABGD] ; this is seen at once if we take P such that APQ is a straight line, and consider the
ANHARMONIC PROPERTIES OF CONICS. 331
ranges made on BG by the two pencils. Hence we have the following converse proposition.
If a point P move so that the cross ratio of the pencil formed hy joining it to four fixed points A, B, C, D, is con- stant ; P will describe a conic passing through A, B, C, D.
Ex. 1. The four extremities of two conjugate chords of a conic subtend a harmonic pencil at any point on the curve.
Let the chords heAC,BD; let E be the pole of BD, and let F be the point of intersection of AG, BD. The four points subtend, at all points on the curve, pencils of equal cross ratio. Take a point indefinitely near to D; then the pencil is D{ABCE}. But the range A, B, C, E is harmonic, which proves, the proposition.
Ex. 2. If two triangles circumscribe a conic, their six angular points are on another conic.
Let ABC, A'B'C be the two triangles. Let B'C cut AB, AC in E', D', and let BC cut A'B', A'C in E, D. Then the ranges made on the four tangents AB, AC, A'B', A'C by the two tangents BC, B'C are equal.
Hence {BCED} = {E'D'B'C}',
.'. A'{BCED} = A{E'D'B'C'\, or A'{BCB'C} = A{BCB'C}, which proves the proposition.
The proposition may also be proved by projecting B, C into the circular points at infinity ; the conic is thus projected into a parabola, of which A is the focus; and it is known that the circle circumscribing A'B'C will pass through A.
323. Def. Kanges and pencils are said to be homo- graphic when every four constituents of the one, and the corresponding four constituents of the other, have equal cross ratios.
Another definition of homographic ranges or pencils is the following : — two ranges or pencils are said to be homo- graphic which are so connected that to each point or line of the one system corresponds one, and only one, point of the other.
To show that this definition of homographic ranges is equivalent to the former, let the distances, measured from fixed points, of any two corresponding points of the two
832 HOMOGRAPHIC RANGES AND PENCILS.
systems he cc, y; then we must have an equation of the form
cy-i d' The proposition follows from the fact that the cross ratio of every four points of the one system, namely
is not altered if we substitute ^ 7 for x, , and similar
ci/j + a *
expressions for x^, x^ and x^.
Ex. 1. The points of intersection of corresponding lines of two homo- graphic pencils describe a conic.
Let P, Q, R, S be four of the points of intersection, and 0, O' the vertices of the pencils.
Then 0 {PQRS} = 0'{PQRS}; therefore [Art. 322] 0, 0', P, Q, R, S are on a conic. But five points are sufficient to determine a conic ; hence the conic through 0, 0' and any three of the intersections will pass through every other intersection.
Ex. 2. The lines joining corresponding points of two homographic ranges envelope a conic.
Let a, &, c, d be any four of the points of one system, and a', 6', c', d' be the corresponding points of the other system. Then aa', bb', cc', dd' are cut by the fixed lines in ranges of equal cross ratio. Hence a conic will touch the fixed lines, and also aa', bb', cc', dd'. But five tangents are sufficient to determine a conic; hence the conic which touches the fixed lines, and three of the lines joining corresponding points of the ranges, will touch all the others.
Ex. 3. Tico angles PAQ, PBQ of constant magnitude move about fixed points A, B, and the point P describes a straight line; shew that Q describes a conic through A, B. [Newton.]
Corresponding to one position of AQ, there is one, and only one, position of BQ. Hence, from Ex. 1, the locus of Q is a conic.
Ex. 4. The three sides of a triangle pass through fixed points, and the extremities of its base lie on two fixed straight lines ; shew that its vertex describes a conic. [Maclaurin.]
EXAMPLES.
333
Let A, B, (7 be tlie three fixed points, and let Oa, Oa' be the two fixed straight lines. Suppose triangles drawn as in the figure.
Then the ranges {abed...} and {a'b'c'd'...} are homographic. fore the peiicils B {abed...} and C {a'b'c'd'... } are homographic.
There-
Ex. 5. If all the sides of a polygon pass through fixed points, and all the angular points but one move on fixed straight lines; the remaining angular point will describe a conic.
Ex. 6. A, A' are fixed points on a conic, and from A and A' pairs of tangents are drawn to any confocal conic, which meet the original conic in C, D and C, D'; shew that the locus of the point of intersection of CD and CD' is a conic.
The tangents from ^ to a confocal are equally inclined to the tangent at A [Art. 228, Cor. 3], therefore the chord CD cuts the tangent at A in some fixed point O [Art. 195, Ex. 2]. So also CI/ passes through a fixed point 0'. Now if we draw any line OCD through 0, one confocal, and only one, will touch the hues AC, AD-, and the tangents from A' to this confocal will determine C" and I/, so that corresponding to any position of OCD there is one, and only one, position of O'CD'. The locus of the intersection is therefore a conic from Ex. 1.
Ex. 7. If ADA', BOB', COC, DOU ... be chords of a conic, andP any point on the curve, then loill the pencils P{ABCD...} and F [A'B'C'D'... \ be homographic.
Project the conic into a circle having 0 for centre.
.334 PROJECTION OF ANGLES.
Ex. 8. If there are two systems of points on a conic which subtend homographic pencils at any point on the curve, the lines joining corre- sponding points of the two systems will envelope a conic having double contact with the original conic.
Let A, B, C, D... , and A', B\ C", D' ... be the two systems of points. Project AA', BE', CC into equal chords of a circle [Art. 319, Ex. 5]; let P, P' be any pair of corresponding points, and 0 any point on the circle; then we have 0{ABCP} = 0{A'B'C'P'}. Hence PP' is equal to AA\ and therefore the envelope of PP' is a concentric circle.
Ex. 9. If a polygon be inscribed in a conic, and all its sides but one pass through fixed points, the envelope of that side icill be a conic.
This follows from Ex. 7 and Ex. 8.
324. Ant/ two lines at right angles to one another, and the lines through their intersection and the circular points at infinity, form a harmonic pencil.
Let the two lines at right angles to one another be xy = 0, then the lines to the circular points at infinity will be given by x^-\-y^ = (). By Ai't. 58 these two pairs of lines are harmonically conjugate.
We may also shew that two lines which are inclined at any constant angle, and the lines to the circular points at infinity, form a pencil of constant cross ratio.
Ex. The locus of the point of intersection of two tan- gents to a conic which divide a given line AB harmonically is a conic through A, B, and the envelope of the chord of contact is a conic which touches the tangents to the original conic from A, B.
Project A, B into the circular points at infinity and the proposition becomes : the locus of the point of inter- section of two tangents to a conic which are at right angles to one another is a circle ; and the envelope of the chord of contact is a confocal conic.
325. The following are additional examples of the methods of reciprocation and projection.
EXAMPLES. 335
Ex. 1. If the sides of a triangle toxicli a conic, and if two of the angular points move on fixed confocal conies, the third angular point will describe a confocal conic.
Let ABC, A'B'C be two indefinitely near positions of the triangle, and let AA', BB', CC produced form the triangle PQR. The six points A, B, C, A', B', C are on a conic [Art. 322, Ex. 2], and this conic will ultimately touch the sides of PQR in the points A, B,C. Hence P^, QB, EG will meet in a point [Art. 186, Ex. 3] ; and it is easily seen that the pencils A{QCPB), B{RAQG], C{PBRA} are harmonic. Now, if A move on a conic confocal to that which AB, AG touch, the tangent at A, that is the line QR, will make equal angles with AB, AG. Hence, since A{QGPB] is harmonic, PA is perpendicular to QR. Similarly, if B move on a confocal, QB is perpendicular to RP. Hence RG must be perpendicular to PQ, and therefore GA, GB make equal angles with PQ; whence it follows that G moves on a confocal conic.
[The proposition can easily be extended. For, let ^ BCD be a quadri- lateral circumscribing a conic, and let A, B, G move on confocals. Let DA, GB meet in E, and AB,DG in F. Then, by considering the triangles ABE, BCF, we see that E and F move on confocals. Hence, by con- sidering the triangle CED, we see that D will move on a confocal.]
If we reciprocate with respect to a focus we obtain the following theorem:
If the angular points of a triangle are on a circle of a co-axial system, and two of the sides touch circles of the system, the third side will touch another circle of the system. [Poncelet's theorem.]
Ex. 2. The six lines joining the angular points of a triangle to the points where the opposite sides are cut by a conic, will touch another conic.
The reciprocal theorem is : —
The six points of intersection of the sides of a triangle with the tangents to a conic draion from the opposite angular points, will lie on another conic.
Project two of the points into the circular points at infinity, then the opposite angular point of the triangle will be projected into a focus, and we have the obvious theorem : —
T\oo lines through a focus of a conic are cut by pairs of tangents parallel to them in four points on a circle,
336 EXAMPLES.
Ex. 3. The following theorems are deducible from one another.
(i) Two lines at right angles to one another are tangents one to each of two confocal conies; shew that the locus of their intersection is a circle^ and that the envelope of the line joining their points of contact is another confocal.
(ii) Two points, one on each of two co-axial circles, subtend a right angle at a limiting point; shew that the envelope of the line joining them is a conic with one focus at the limiting point, and that the locus of the in- tersection of the tangents at the points is a co-axial circle.
(iii) Two lines ichich are tangents one to each of two conies, cut a diagonal of their' circumscribing quadrilateral hainnonically ; shew that the locus of the intersection of the lines is a conic through the extremities of that diagonal, and that the envelope of the line joining the points of contact is a conic inscribed in the same quadrilateral.
(iv) AOB, COD are common chords of two conies, andP, Q are points, one oh each conic, such thatO{APBQ\ is Tiarmonic; shew that the envelope of the line PQ is a conic touching AB, CD, and that tlie tangents at P, Q meet on a conic through A, B, C, D.
(v) If two pointa be taken, one on each of two circles, equidistant from their radical axis, the envelope of tlie line joining them is a parabola which touches tlie radical axis, and the locus of the intersection of the tangents at the points is a circle through their common points.
Examples on Chapter XIV.
1. Shew that an hyperbola is its own reciprocal with respect to the conjugate hyperbola.
2. Shew that a system of conies through four fixed points can be reciprocated into concentric conies.
3. Shew that four conies can be described having a common focus and passing through three given points, and that the latus rectum of one of these is equal to the sum of the latera recta of the other three. Shew also that their directrices meet two and two on the sides of the triangle.
4. If each of two conies be reciprocated with respect to the other ; shew that the two conies and the two reciprocals have a common self-conjugate triangle.
EXAMPLES. 337
5. Two conies L^ arxd L^ are reciprocals with respect to a conic TJ. If J/j be the reciprocal of L^ with respect to X^, and Jfg be the reciprocal of L^ with respect to Z, ; shew that M^ and J/g are reciprocals with respect to TJ.
6. If two pairs of conjugate rays of a pencil in involution be at right angles, every pair will be at right angles.
7. If two pairs of points in an involution have the same point of bisection, every pair will have the same point of bisec- tion. Where is the centre of the involution %
8. The pairs of tangents from any point to a system of conies which touch four fixed straight lines form a pencil in involution. Hence shew that the director circles of the system have a common radical axis.
9. Two circles and their centres of similitude subtend a pencil in involution at any point.
10. If two finite lines be divided into the same number of parts, the lines joining corresponding points will envelope a parabola.
11. If P, P' be corresponding points of two homographic ranges on the lines OA^ OA', and the parallelogram POP' Q be completed j shew that the locus of ^ is a conic.
12. Three conies have two points common ; shew that the three lines joining their other intersections two and two meet in a point, and that any line through that point is cut by the conies in six points in involution.
13. Shew that, if the three points of intersection of corre- sponding sides of two triangles lie on a straight line, the two triangles can both be projected into equilateral triangles.
14. Shew that any three angles may be projected into right angles.
15. A, B, C are three fixed points on a conic j find geometrically a point on the curve at which AB, PC subtend equal angles.
16. Through a fixed point 0 any line is drawn cutting the sides of a given triangle in A', P\ C respectively, and P is the point on the line such that [A'B'C'P] is harmonic ; shew that the locus of P is a conic.
S. C. S. 22
838 EXAMPLES.
17. When four conies pass through four given points, the pencil, formed by the polars of any point with respect to them, is of constant cross ratio.
18. If two angles, each, of constant magnitude, turn about their vertices, in such a manner that the point of intersection of two of their sides is on a conic through the vertices, the other two sides will intersect on a second conic through their vertices.
19. If all the angular points of a polygon move on fixed straight lines, and all the sides but one turn about fixed points, the free side of the polygon will envelope a conic.
20. If a polygon be circumscribed to a conic, and all its angular points but one lie on fixed straight lines, the locus of that angular point will be a conic.
APPENDIX.
ANSWERS Sec. TO THE EXAMPLES.
CHAPTER II.
3. Ans. i)o(^-a)"-i'i(^-«r"My-&)+i'2(^-ar"M2/-^r--- + {- l)^Pn{y - b)^ = 0. 4. The lines make equal angles with one another. 5. Take OA, OB for axes, and let OA, OB, OP, OQ be a, b, h, k respectively. Since AP=c .BQ, we have h-a=c{k-h). If {x,y) be middle point of PQ, 2x = h, 2y — k; whence required locus is 2x~a = c{2y- b). 7. Take the fixed lines for axes and let P be {x, y) and Q be {x', y'). Then x' = x + yco8o}, y'^y + xcosu. Find x and y in terms of a^ and y', and substitute in the equation of the locus of P. 8. Use polar co-ordinates with O for pole. 10. The equations of AB, AD, BC, CD Aie d = 0, d-a, rsin (^-a) + asina=0, and r sin 0 = 6 sin a; where a, 6 are the lengths oi AB and AJ), and a is the angle BAD.
The equation of AC is 0 = tan-i ; — —, and of BD is rasing
a + b cos a
-a6sina + &rsin(a-0)=O. 14. Ans. 72/-3a;-19=0, 7a: + 3r/ - 33 = 0,
7y- 305 + 10 = 0, and 7x + 3?/-4 = 0. 15. If the base be taken as axis
of X, the sum of the positive angles the sides make with it is constant.
IP rri, A- f 4.1 1 • y^ + (x-a)[x-b) y^ + {x-c){x-d)
16. The equation of the locus is 5^ '-^ -' = ^^ ^ .
a-b c-d
The points are on the axis of x and a, h, c, d are their distances from the
origin. 18. The equation of the locus is (a - a')xy + l{x~ a) {x - a') = 0, where
AB is axis of x, the other given line the axis of y, and OA=a, OB=a',
and I the intercept on axis of y. 21. The bisectors of the angles between
the two pairs of straight lines coincide [Art. 39]. 22. Ans. {db' - a'b)^
= 4i(ha'- h'a) [h'b - hb'). 23. This is reduced to the preceding by means
of question 2. 26. The result easily follows from the polar form of the
equation, viz. 7/1 tan 30 + 1 = 0. 29. If the straight lines be y^m^x,
j,=,V, and y^m^; we have ie= v^Zj:^^^_^i^^^ . But
-m^x)^ {y - m^f {y -rn^y
22-
340 APPENDIX.
ay^ +by^x + cyx^ + cb^ = a{y- m-^x) {y - m^) {y - m^) , and { 1 + rrij^) (1 + m^^) (1 + m^^) = 1 + (wii + mj + 7713)2 - 2 (WiWia + m^m^ + m^rn^ + {m^m^ + m^m^
53 c c^ dh d' + W3Wii)»-27nim2W3(mi+m2+m3) + ?ni2m2W=l+7i -2- + -a-2-2+;;25
Of Cb CL €L a
whence the result. 30. The equation of any pair of perpendicular lines is x^ + \xy -y^=0. Hence given equation must be equivalent to {Ex + Fy ) {x'^ + \xy-y^)=0, BO that E=A, F= -D, F+\E = SB, and \F-E=ZC.
33. The lines are ax^ + 2hxy + by^-^-,{a'x'^ + 2h'xy + bY)=0 [Art. 38].
34. Let A, B, 0 be (Oj, 61), (a^, 63) ^^^ (<»3> ^3); and A', B', C be (oj, jSj), (ttj, /Sg), (ttg, ^3). The equations of the three perpendiculars from ^', B', C on the sides of ABC are x (a^ -a^+y {\ - h^ - a^ {a^ - a^ - jSj {h^ - 63) = 0 (1), « K - «i) + 2/ (&3 ~h)-<h {«3 - <h)-§2 {h - &i) = 0 (2), and a; (o^ - ag) + 2/ (&i - 62) - tta («i - «2) - i^s (&i - 62) = 0 (3). If (1), (2), (3) meet in a point the sum of the constants is zero, and this sum can be written in the symmetrical form a-^a^ - a^a^ + a2a3 - ^302 + a^a.-^ — a^a^ + 61^2 ' ^ftPi + &2/33-63^2 + &3i8i-&i)33 = 0.
CHAPTEB IV.
4. The locus is {1-n^) {x^ + y^ + a^)-2a{l + n^)x = 0, where (a, 0), (-a, 0) are the two points Ay B. The common radical axis is .r = 0. 6. Aiis.X'-\-y^ + 2dx + 2ey+f{Ax + By) = 0. 7. Ans.2x^+2y^ + 2x+6y + l=0.
8. See Art. 38. 9 and 10. Substitute — for r in the polar equation of the
r
line or of the circle. 12. If a common tangent, PQ, of two of the circles
cut the radical axis in 0, the tangents from 0 to all the other circles of
the system, including the limiting circles, will be equal to OP; therefore
the limiting points are on a circle on PQ as diameter. 13. If one circle
is within the other, (1) the radical axis must cut in imaginary points,
therefore 6 is positive; (2) the centres must be on the same side of the
radical axis, therefore a and a' have the same sign. 15. See Art. 86.
19. We may take
a;cosa + i/sina-a«:0, .Tcos ( a+ — j+7/ sin ( a+ — J -a = 0, &c.
for the equations of the sides. The sum of the squares of the perpen- diculars from (x, y) is sum of squares of left sides ; and in this sum the
coefficients of x^ and y^ are equal, since cos 2a + cos 2 f a H j + .... =0;
also the coefficient of xy is zero, since sin 2a + sin 2 ( a + — J + .... =0.
ANSWERS TO THE EXAMPLES. 341
20. - + - = 2, (h, k) being the point through which PQ passes. 21. If P X y
be any point on the circle, and A, B the ends of a diameter, PA'^+PB'^=AB^ ;
express this in polar co-ordinates. 22. Eliminate d. Condition for
tangency is j? = 2a cos^ | or j? = - 2a sin^ ^ . 23. a; = - , t/ = - j . 24. Two
circles. 25. The whole lengths of the lines, from the points of contact to their intersection, are equal to one another. 26. The given lines must intersect on the radical axis of the circles. 27. If given points are (±a, 0), and tangents are parallel to y=xiQXid, the equation of the locus is y^ + 2xy cot 6 -x^ + a^ — d. 29. For straight lines, A + B + G = Q. 31. Any circle through (±a, 0) is x'^ + y^-2hy -a^ = 0. The orthogonal circles are x^-\-y^ -1cx + a^=.0. 33. Take x'^ + y^- 2ax = 0, x^ + y^-2hy = 0 for the equations of the circles. 35. The equa- tion of the locus is (62 + c^) (a;2 + 2/2 + ^S) _ ^^abcy = [x^ + y^- a^f-. The bisectors touch the circles a;2 + (y±a)2 = |(6±c)2. 36. The centre of the required circle must be the radical centre of the three escribed circles. The equation of circle touching BC, and AB, AG produced is x^ + y^ + 2xyG08A -2s {x+y) + s^ = 0 (i), AB, ^C being axes. The radical
"7* ?y ij* ?y
centre of the escribed circle is given by j- r=i. - H — -^ =h, its co-
a+c be a+b
ordinates are therefore -~ — - and — ^-j — - . The radius required is equal
to the tangent to one of the circles from the radical centre, and this is found by substituting the co-ordinates in (i), 37. Let the centres of the circles be {x', y'), {x'\ y"), the fixed points (±a, 0), and the point of contact {x, y) . Then we have (i) {3i-af + y'^ = c\ (ii) {x" + af + ?/"2 = c\ (iii) {x'-x!y + {y'-y'y = ic^. Also 2x=x' + x", and 2y = y' + y". From (i) and (ii) {x' ~ x") {x' + x") - 2a {x' + x") + {y' - y") {y' + y") = 0, i.e. x{xf- x") - 2ax + y [if - y") = 0. This with (iii) gives us {y' - y") and {x' - x"). Then, taking (iii) from twice the sum of (i) and (ii), we have (x' + aj")2 -}-(?/' 4- y")^ + 4a2 - 4a (a/ - ic") = 0 ; whence the required locus.
CHAPTER V.
(«+l)2
4. The parabola is y^= -ax, where 1 : n is the given ratio.
n
5. (i) a straight line through the vertex, (ii) the curve y^ = nx'^ + 2ax.
6 y^ = x^ + 6aa; + a2. 10. The chord of contact of tangents from ( - 4a, k)
is 1/ ft = 2a (a; -4a). The equation of the lines joining vertex to points
342 APPENDIX.
of contact is [Art. 38] y"^- 4aa; ^^ =0, or ^"-3:2+ 2-^y=0.
,.'2 _ 4^^'
11. TN= "„ ,., , ^, , and T3P = y"^ + 4a2 ; where {x\ y') is T. 12. Let
the axis of x be midway between the axes of the parabolas, then their equations will be {y -hY = ^ax, {y+b)^ = ^ax. If 2/ = 17 cut the curves in {xy, tl) and (aig, ri) respectively, we have rp-\-h^ = 2a{,x-i^-\-x^. Hence, if (^, 77) be the middle point of intercept, ift^ + h^ = 4ia^. 15. The chord whose middle point is {x', y') is parallel to the polar of (x', y'); its equation therefore is {y - y') y' = 2a {x - x'). If the chord pass through the fixed point (h, k), we have (k-y'}y'=:2a{h-x'). Hence the required
locus is the parabola y {y - k) =2a {x - h). 25. Let y = miX-\ — ...(i), y=m^A — ...(ii), y = 7n^x+—...{m.), and y=m^x-\ — ...(iv),be theequa-
70,2 W3 m^
tions of the four tangents. The ordinate of the point of intersection of (i) and (ii) is a ( — + — ) , and the ordinate of the point of intersection of
(iii) and (iv) is a{ 1 ) ; hence the ordinate of the middle point of
these intersections is -( — H + — + — |. The symmetry of this
2 \mi 7/12 ^3 ^4/
result shews that the ordinate is the same for the middle point of the
other two diagonals. 27. If the fixed line and the two tangents
make angles a, ^j, O^ with the axis, we have 2a = 0y^ + d,2. And if
{x', y') be the point of intersection of the tangents, tan^j and tan6>g
are the roots of y'=mx'+ - . "We therefore have tan 2a= , : which m x - a
shews that the intersection of the tangents is on a fixed straight Hne ;
therefore, &e. 33. At points common to y^ — 4cLe=0 and any circle
x'^ + y^ + 2gx + 2fy + c=0, we have ^-^ + y^ + 2g ^ +2fy + c = 0. The co-
efiicient of y^ is zero, hence 2/i + 1/2 + 2/3 + !/4 = 0. If therefore the normals at 2/1, 2/2 » 2/3 ^^6et in a point, 2/4 is zero; for we know that 2/1 + 2/2 + 1/3 = 0 [Art. 106]. 38. The normal at {x', y') is 2a {y - y') ^y'{x-x') = 0. If
this pass through {h, k), we have 2ak + y' (h~x' -2a) = Q, whence 4a2/c2 = Aax' {h -x' - 2af. This gives a cubic equation for x' from which we have «' + x" + a/" — 2'h- 4a, or (x' + a) + {x" + a) + {x'" + a) + a = 2/j ; therefore, &c. 41. Take for axes the tangent parallel to the given lines and the diameter through its point of contact. 43. The line is a; = 2a + c. 44. The ordinates of the normals which meet in {h, k) are given by 2a (y - it) +
ANSWERS TO THE EXAMPLES. 343
y {x- h) = 0. If {hy k) be on the curve, we have 2a (y - k) + j— {y^ - k-) = 0.
Hence the ordinates different from k are given by ?/(// + ^•) + 8a2=:0 ; go that yiyi=^a^. The equation of the chord is y {yi+y^-^ax-y^y^—0^
hence this cuts the axis where a; = - ^- =- 2a. 47. Let yi,y2,yzt y^
be the ordinates of the points A, B, C, D ; and let AB, BC, CA, AD make angles ^j , 6^, 6.^, 0^ with the axis. Then [Art. 102], ?/i + 1/2 = 4a cot O^ , and so for the rest. Hence cot ^^ + cot ^3 = cot ^2 + ^®* ^4 5 which shews that if three of the angles are constant, the fourth also is constant. 50. Let P, Q, R, S be (x^, y^) &c. The equation of the circle on PQ as diameter is (y - y-^) {y-y^) + {x- Xi) {x - x^) = 0. Where this meets y^ = 4^ax,
we have (y - y^) [y -y.^) + ^^ {y^ - y^^) {1/ - y^^) = 0. Hence 2/3 , 2/4 are the
roots of IQa^ + (y + yi) {y + y.2) = 0, so that 2/32/4=2/12/2 + l^a^. But PQ, RS
cut the axis at points whose abscissae are - ^-^ and - -^- , hence
the difference of these abscissae is 4a.
CHAPTEB VI.
4. The equation of a line through the middle point of a chord per- pendicular to the chord can be written down by assuming Art. 114 (iii).
15. The ellipse is -' + ^3 = f ^^^^ ^. 18. Use y=mx+ s/{ahn^ + J2). (t 0 ya" -\- 0 J
20. Use eccentric angles. 21. The line y = m{x-ae) cuts the ellipse
where — „ + — ^^^ — - = 1. Put x = hlae+-\-p, and shew that the
product of the roots of the quadratic in p is independent of m. 27. If
/ 1 - e2\ P be (x', y'), the point of intersection of QH and RS is ( - x', - y' ^ J .
30. The semi-axes of the locus of P are the semi-sum and semi- difference of the radii of the circles. 36. The chord which has {x\ y') for middle point is parallel to the polar of {x\ y'), its equation is therefore
{x - x') —, + (y-y')^^ = 0. Hence, if the chord pass through a fixed point
{h, k), the middle point is on the ellipse (h—x) -^ + (^ - 2/) -^ = 0. 37. Let
P be («', y'), and let the chord make an angle 0 with the major axis of the ellipse. The co-ordinates of the point on the chord at a distance r from
844 APPENDIX.
P are .r' + rcos^, and y' + r sin 6; substitute these co-ordinates in the equation of the ellipse for FQ, and in the equation - + ^ = ^ for PR ;
-j + ^ -1 j
40. Let a, /9, 7, 5 be the co-ordinates of the angular points A, B, C, D
of the quadrilateral ; then, since A B, BC, CD are parallel to three fixed
straight lines, we have (a + /3), 03 + 7) and (7 + 5) constant; therefore
(a -I- 5) is constant. 43. Let the co-ordinates of Q be a cos ^ and 6 sin ^,
then quadrilateral OPCQ = 2 . triangle OCQ = hb sin d - ka cos 6 = A, sup-
A h k
pose. Therefore -r = - sin^ - - cos d (1). But {h, k) is on the tangent
h k
at <3; therefore -cos ^ + - sin ^ = 1 (2). From (1) and (2) we have
-^ = ~2 + Ta ~ ■'•• "^^^ *'^^ °^ ^^^ triangle P(7Q can be readily deduced
from that of the quadrilateral. 50. Ans. 2 {h-y'^ -{â– a^x^f = {a^ - h^)'^
{a^x^ - b^y-)^. 63. If 0 be the eccentric angle of P, the co-ordinates of Q are {a + b)co8^ and (a + 6) sin 0, or (a-&)cos0, (6-a)sin0, ac- cording as PQ is measured along the normal outwards or inwards, 55. Let T be {x', y'). In the quadratic equation giving the abscissas of
points where ^-fi^ = l cuts the ellipse, substitute --- for.T [Art. 110];
the product of the roots of the equation in r will be equal to SP .SQ.
CHAPTEK VII.
3. An hyperbola. 4. An hyperbola. 5. A rectangular hyperbola. 19. 2y + 3x + 4t = 0. 20. x-2 = 0, y -S = 0, xy -3x-2y + 12 = 0. 26. The lines joining {x'y') to the two fixed points (± a, 0) are {yx' — xy')^=a-{y-y'f. These are parallel to y^{x'^ — a-)-2x'y'xy + y''x^ = 0, the bisectors of
which are -— ~ — :; + -^-,=0. Since these bisectors are fixed lines, we
y ^-x-' + a-' xy
have ~ r~, = const.
x'y'
ANSWERS TO THE EXAMPLES. S45
CHAPTER Vni.
4. If a, /3, 7 be the vectorial angles of A, B, G respectively,
8A = , a.ndLSA' = , &c. 5. As in Art. 165 (5), the
2 cos*^ ^ 2 cos ^ cos ^
perpendicular on the tangent at a makes with the axis an angle
tan-i ^^— . Therefore, &c. 7. See Art. 165 (3). 10. If the conies e + cosa ^
are - = 1 + e cos ^, and- = 1 + e'cos (^ - a), the common chords are - - e cos Q
r |. \ ' J.
= ^\~- e' cos (6- a)\ . 13. If the conies are - = 1 + e cos 6, and ~
(r ^ ') r r
IzizV
= l + e'cos(^-a), the common chords are — =e cos d±e'cos{d - a).
IzhV e'
These touch respectively the conies - -=:1±- cos 6. 16. If d be the
distance of the focus from the directrix, the conies will be — = 1 + e cos d,
r
e'd and — = l + e' cos {6 - a). If the conies touch one another at some point
/3, the equations — — ecos 0+ cos {d- p), and —i- = e' cos {d - a) + cos {6 - ^)
will represent the same straight line. Write the equations in the
„ d ^ /. coSi3\ . . sinjS ^ d ^f , cos j3\
forms = cos ^ 1 H ) + sm ^ , and - = cos 6 cos a H ,— )
r \ ^ J ^ *â– V ^ J
+ sin^ I sin a-\ ^^ J ; equate the coefficients of cos 6, and of sin d, and
eliminate /3. 17. Let the equation of the circle be r=a cos [6 - a), and
the equation of the conic - = 1 + <? cos 6. Eliminate d, and we obtain a r
biquadratic for r.
CHAPTER IX.
7. Ans. X = l. 8. Ans. 10a:2 + 21a;2/ + V-41a;-39?/ + 4 = 0. 9. Ans. 3a;2-2a;?/-5?/2+7a;-9^ + 2 = 0, and 3«2 - 2a;?/ - 5?/2 + 7a; - 9?/ + 20 = 0. 10. Ans. 6x2 -7a;2/- 3^/2 -2a; -8?/ -4 = 0, and ^x^-lxij -Zi/-2x-Qy-2 = 0. 14. Take 0 for origin, and the axis of x through the centre of
the circle. The equation of the circle will her = d cos d (1) ; the equation Qf the conic ax"^ + 'ihxy + hy^ + 2^a; + 2fy + c = 0, or in polars ar^ cos^ Q
34 G APPENDIX.
+ 2;<v2cos<?sin^ + 6r2sin2^ + 2f7rcos^ + 2/rsin^ + c = 0 (2). Eliminate 6 from (1) and (2), then we obtain an equation in r the product of the four
roots of which will be ^ ^ — jvg • Since the origin is fixed, c is
constant ; and [a - 6)^ + 4^^ ig constant from Ex. 11.
CHAPTER X.
3. To find the fixed point in Ex. 1, take OP, OQ parallel to the axes ; then PQ, is a diameter, and CO, PQ make equal angles with the axis. Hence the co-ordinates of the point can be found referred to the centre and axes of the conic. The fixed point in Ex. 2 is the point where the tangent at 0 is met by the tangent at the other extremity of the normal through 0, as is seen by taking OP, OQ indefinitely near to the normal. For locus see Art. 138 (4). 7. Take 0 for origin and the chords for
axes. We have to prove that — — is independent of the direction of the
axes. 13. In the parabola y'y" is constant. 20. Take 0 for origin, the chord and its conjugate for axes; then the equation of the curve will be ax^ + by^ + 2fy + c = 0. Tangents from {x', y') are given by <p {x, y) <p {x\ t/) - {axx' + byi/+f{y + y') + c}^ = 0; in this put ?/ = 0 ; then the coefficient of X will be zero if fy' + c = 0, that is if {x', y') be on the polar of 0. Or, let the tangents at P, Q meet in K ; then KL, the polar of 0, is parallel to AB ; and if QOP meet the polar of 0 in L, {QOPL) is harmonic. Hence {TOS oo} is harmonic, and therefore TO — OS. 22. (i) a conic ; (ii) a straight line. 25. A curve of the fourth degree. 28. Corre-
sponding to any point T on the tangent at P there is oiie point T such that T, T' are equidistant from the centre, and there is one inter- section of the tangents at T, T'; hence every tangent to the ellipse cuts the locus in one and only one point : the locus is therefore a straight line. If T, 1" are on the director circle, the tangents from T, T are parallel ; therefore the direction of the point at infinity on the locus is perpendicular to the tangent at P; also when T, T' are both at infinity, the tangents from T, T are parallel to the tangent at P, and therefore intersect at the extremity of the diameter through P, which proves the proposition. 37. The centre of the conic is
given. Hence, if P be the given point, P', the other extremity of the diameter through P, is on all the conies. The locus is such that SP . SP' is constant; this curve is called a lemniscate. 40. Let S, S' be the foci, S being given, C the centre, P the given point, and 0 the middle point of SP, Then CD'^ = SP , 8^=^4.80 , OC. This proves that
ANSWERS TO THE EXAMPLES. 347
the locus of i) is a parabola, since CD and 00 are drawn in fixed directions, and SO is fixed. 43. Let the variable ellipse touch at P, and let the tangents at 5f, P meet in T. OT bisects SP in V, and is therefore parallel to S'P, so that CT and SP make equal angles with the tangent at P; hence VT=VP=VS\ therefore STP is a right angle, and CT is the radius of the director-circle of the variable ellipse. Hence, since CT=l {SP + S'P)= constant, the question is reduced to 37. 44. This follows from Ex. 23, Chapter vii. 47. {PGOG'} is harmonic, and GCG' is a right angle ; therefore CP and CO make equal angles with CG. Then see solution of 2. 53. Ans. c=±a6. 54. Let the conic which goes through A, B, C,D, E cut the circle ABE in G\ then, AB and CD make equal angles with the axes, and so also do AB and EG, hence EG is parallel to CD, so that G and F are coincident. The direction of the axes is known, we have therefore only to find the centre. If F, V are the middle points of CD and EF respectively, VV is a diameter. Draw a circle through i), C, E: if this cut the conic in a fourth point H, EH and CD make equal angles with the axes of the conic ; therefore EH is parallel to AB ; hence the line through the middle points of AB and EH is another diameter. Thus the centre is found. 55. The six points are always on a conic, and the conic is (axV + ly'y" - 1) {ax^ + hy"^ - 1) - [axx' -f hyy' - 1) [axx" + hyy" - 1) = 0
[see Ex. 3, Art. 187]. The conditions for a circle are x'x" - y'y" = — - (1),
(1 1\2 - - - J ,
that is CP . CP'~CS^, where C is the centre and «S^ is a focus; also from (2) CP and CP' make equal angles with the axis of x; and (1) and (2) shew that P, P' are on different sides of the transverse axis. When the curve is a parabola P, P' are on a line through the focus, and equi- distant from the focus. 58. The chord of ax^ + hy^ -1 = 0 which has {x', y') for middle point is parallel to the polar of [x', y') and its equation is {x - scf) ax' + {y - y') by' — O. The line through {x', y') perpendicular to the
chord must pass through 0 (/, g) ; hence we have —-r = ^t^t" » so that
{x', y') is on a rectangular hyperbola. 59. Any conic of the system is given by ax^ + hy^-l-\{{x-af-Â¥{y-^Y-c^ = Q, where (a, /3) is the point 0. Find the centre, and eliminate X. 63. If the normal
to ax2 + 6t/2 _ 1 = 0 at P (x',y') pass through 0 (/, g) we have*^^-^ = ~-~ »
ax oy
or f by' - agx' + (a - 6) x^y' =0(1), We have to shew that ^-^^ + -^ = 0,
X y
348 APPENDIX.
or xy' + yx' -2x'y' = Q, vdll go through the same point if {x\y') is any one of the four points of intersection of (1) and the conic. The point
is — = ^ =z . 73. A conic. 75. The four points are
Jh - ag b-a
— ,-) &c., where {x'y') &c. are the feet of the normals. Now, if
(/, g) be the point at which the normals meet, -y j = a'^-b^.
X y
fc^ b^\ Hence {—,-?) is on the straight line/r - gy = a^- b^, and so also are the
\« y J other three points. 83. If y = m{x-ae) be the chord, the cu-cle is
a;2-a2 + 3^2 + 2mae?/-m262=:0,or^ + |^-l-^3(ae2/- w62)2 = 0. 96. If
xy^<? be the equation of the hyperbola, and {x^, y-^ &c. be the four
jK>ints.a»d(a,;S,beP; then ^ .Pa. '' '^- "^'^^^^^ '^^-^' â–
CHAPTEK XI.
3. Let the equation of the conic which passes through O be ax^ + 'ihxy + by'^ + 2fy = 0, the tangent and normal at 0 being axes. If a'x^ + 2h'xy + b'y^ + 2g'x+2f'y +c' = 0 be the equation of another conic, all the conies through their common points are included in ax^ + 2hxy + by^ + 2fy
+ X {a'x^ + 21i'xy + b'y^+ 2g'x -h 2fy + c') = 0. Put y=0, then — + —
x^ x^
2a' = — 7" , and therefore is independent of X. 5. The axes of the para- bolas are always parallel to conjugate diameters [Art. 207]. Now in a given ellipse the acute angle between two conjugate diameters is least when they are the equi-conjugates ; and in different ellipses the angle between the equi-conjugates is greatest in that which has the least eccentricity. Hence if a pair of conjugate diameters are known, the conic has the least eccentricity when they are the equi-conjugates. 6. If TQ, TQ' be the tangents, and Fbe the middle point of QQ', TV and QQ'oxe parallel to conjugate diameters. See solution to 5. 16. Use the result of Ex. 2,
Art. 219. 21. A circle. 24. tan2-=: --^, where ^is the angle between
the tangents. 25. The result follows from Art. 229 and Art. 186, Cor. 1. 28. Art. 227. 35. If TO' be the other bisector of the angle QTP, then T {QOPO'] is harmonic, and therefore TO' is the polar of 0. Let BOH' cut TO' in K, then T {BOR'K} is harmonic, and OTR is a right
ANSWERS TO THE EXAMPLES. 349
angle; hence RT, R'T make equal angles with OT. 37. and 38. Use 'If a circle cut a parabola in four points the sum of the distances of those points from the axis of the parabola is zero.' 42. Shew that the conic, with respect to which the triangle formed by a: = 0, y = 0, and lx+ mr/ + 1 = 0 is self-polar, is ax^-\-2lmxtj + hy'^ + 2lx + 2my + l — 0. 50. Let the hyperbola be 2xy = c, and the first circle x^ + y^ + 2gx + 2fy = 0. Let {x^, y^) &c. be the four points. The equation of the second circle is x^-\-y^ + 2xx^ + 2ijy^ ~ 0. The point of intersection of the tangents at I>* , G
^ ^ ' • this is on the second circle if c^ [x^ + 3c.^ + ^4)
is (i
+ 4x^^xJ^X2X^ + x^x^ + X4X2) = ^i^)' Now the equation giving the abscissas of ^, J5, C, D is Ax:^ + SgT'' + 4:fcx + c^ = 0. Hence 4:Xt^x.2X^x^ = c^, and a?! (acg + x^ + x^) + x^^ + x^x^ + x^x^ = 0 ; and these shew that (1) is true.
CHAPTER XII.
1. A parabola. 3. An hyperbola. 4. (1) A similar ellipse. (2)
An ellipse. 12. A common chord, which is not a diameter, subtends a
right angle at the centre. The envelope is a circle. 16. If the
conic is dx^ + hy^^l, and c the radius of the circle, the envelope is
dbc^ 1 1
ax^ + Sw2= . The envelope is the original conic if c^^- + - ; that is,
^ a + 6 ^ "^ ah'
if the circle is the director-circle of the conic. 20. See Art. 197. 25.
The equation of the envelope is xy= d=4a6. 26. If the original conic
is ^-f 1-^=1, the envelope is ^ -f |5-f 2 ~ = 0. 27. Take the fixed
a^ 0- ^ a^ 0^ a
point for origin, and let the lines be {x - a) {x -a') = 0 ; then the envelope
is {a-ayy^=4:aa'{x-a){x-a'). 31. Take the given diameters for
axes, and let the conic be asi^ + 2hxy + by^ -1 = 0; then the envelope is
4:{ax + hy){hx + by) = -— -. 38. Let the equation of the conic be
ax^ + l)y^ = l, and let 0 be (a, /3). Transfer the origin to 0, and let lx + my + 1 = 0 be the equation of PQ, one of the chords. Write down the equation of OP, OQ [Art. 38]; then the condition of perpendicularity gives the tangential equation, viz. {l^+m!^) {aa^ + b^'^-l)-2aal-2b^vi
+ a + b = 0. One focus is (0, 0), the centre is ( r, —,] , and
\ a + b a + bj'
the other focus is ( , , ^V 1 • If a : ^ is constant, the envelopes
V a + b' a + b) ^
are confocal. If the given conic is a rectangular hyperbola a + 6 = 0, and
the envelope is a parabola.
350 APPENDIX.
N.B. The following are important special forms of the tangential equation 0 (I, m) = 0.
(i) If c = 0, the conic is a parabola.
(ii) If a = 6 = 0, the conic touches the axes.
(iii) If a = 6, and /t = 2aco8 w the origin is a focus.
CHAPTER XIII.
Ans.
abc a\ /S', y' 8A« a", /3", y" a'", j8"', y"
3. The four points of intersection of any two of the conies are of the form ±/, ±gr, ± /j. The conic Ma* + v/32 + W7' = 0 will pass through the points (±/, ±^, ±h), and (±/', J^g', i ^') it up + vg^ + wh^ = 0, and uf'^ + vg'^ + ich'^ = 0. 6. Let the lines be laJ^mp±ny = 0; then the
two points on the diagonal a = 0 are given by nv^fi^ + 2X/37 + w^y" — 0, a = 0. The other pairs are 7*372 + 2fiya + Pa^ = 0, p = 0; and Pa^ + 2vap + m2/3» = 0, 7 = 0. These are all on the conic Pa? + m^^ + n^y"^ + 2X/S7 + 2fiya + 2va/S = 0. 7. The perpendicular distances of (a, /3, 7) from the three sides are
- («a - 6j3 - cy), &c. Hence the equation required is
'^ + -^^=0.
bp + cy-aa cy + aa-bp aa + bp-cy 8. The equation of the circle is of the form apy + hya + ca^ + \ (aa + 6/S + cyy = 0. If this cut BC in P, P\ then BP . BP' = r''-R'^. 9. The point which is at a distance p from (ap, p^, y^), on a line parallel to iJC is (ttQ, ^Q + psin C, 7^ -psin B). If this point be on the conic, we have
^^^- -ZsinBsinC
Hence we have
2 «.3
-S- = -i
a & c I m n [If the conic were given by the general equation we should have
„ _ = 0 (ao> /3o> 7o)
^^ ^2 V sin2 C + WJ sin^i? - 2 m' sin ^ sin C *
Hence we find at once the conditions for a circle, viz. vc^-\-%ch'^-2ii'bc
= similar expressions. ]
ANSWERS TO THE EXAMPLES. 351
11. If P be (/, g, h), K, L, M are on the line ^ + ^ + J = 0. If P be on la + mp + ny = 0 KLM touches J la+ Jm^+ Jny = 0 &c. 12, If O be (/, g, h) and 0' be (/', g', h') then Z is given by Jfrj^j^rzYh) ~ ^ - "^ If 0, 0' are on the fixed conic
gg'ihf-h'f) hh'ifg'-fg) 'KPy + /Mya + pap = 0, Z is the fixed point (X, /x, v).
(Examples 11 and 12 are taken from an interesting paper by Mr A. Martin published in the Messenger of Mathematics, Vol. IV.) 18. If ua^ + v^^ + wy^ = 0, be a parabola it will touch the line at infinity, and therefore all the four lines given by aai6/3±C7 = 0. 20. If
(a', p', y) be one focus, the other will be (-,,-,, -) ; write down the con-
\a p y J
dition that the fixed point (/, g, h) may be on the line joining these two
points. 34. Let (/, g, h) be the point of intersection of ^^', BB', CC,
then A' is (/', g, h), B' is (/, g', h), and C is (/, g, h'). BC, CP' intersect
in A" where - = ^= V, . Hence the equation of ^' J." is f 9 ^
a /3 7 =0.
r 9 n
f 9' h'
It is clear that A' A", and also the other two diagonals P' P", C" C", of the hexagon formed by the six lines, pass through the point (/+/', 9 +9', h + h'). Hence by Brianchon's Theorem the hexagon circumscribes a conic. 40. Consider any two of the conies, and draw their fourth common tangent. Then, the radical axis of their director-circles is the directrix of the parabola touching the four lines [Art. 299 (5)]; the radical axis therefore [Art. 308, Ex. 4] passes through. the orthocentre of the original triangle. Then, since the director-circles are equal, it follows that the centres of any two of the conies, and therefore the centres of all the conies, are equidistant from the orthocentre of the triangle. 41. The centre of the circle with respect to which the triangle is self-polar is the orthocentre. Hence, from 40, the theorem will be true for all conies whose director-circles are equal, if it be true for any one of them. Let ^PC be the triangle, and 0 the orthocentre, and let OA cut BG in A\ Then, if P be any point on BC, the line ^P is a limiting form of an in- scribed conic, and the circle on ^P as diameter is its director-circle ; also OA . OA' is equal to the square of the tangent to this circle from 0, and OA . OA' is equal to the square of the radius of the self -polar circle, hence
352 APPENDIX.
the self-polar circle cuts the director circle at right angles. This proves the proposition, since P is any point on BG. 50. The equations of the tangents from the angular points of the fundamental triangle are Vz^+Wy^ - 2V'yz-=0, &c. Hence the six points are on the conic
VWx'' + WUy^ + IJVz^ - 2 UU'yz - 2 VV'zx - 2WWxy = 0, This intersects
l/'^x^ + Vy + W'z'^ -2V' W'yz - 2 W ITzx -IV Vxy = 0 in the same four points as
(FPf-f7'2)ic2-i- +2{V'W-UU')yz + =0.
But this latter conic is the original conic, since VW- V^^uA, &c.
THE END.
CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.