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minimi
6000471 18P
I1$L I ■
Quf.
4
f
A TEEATISE ON
PLANE CO-ORDINATE GEOMETEY.
/tli e.
f*
A TREATISE ON
?im- gmimmii eraum
AS APPLIED TO THE STRAIGHT LINE
AND THE
cevie lienors.
Whfy ^nmermts (foarajrlw.
Bt I. T@©S¥NfBJl, M.A.
FELLOW AND PRINCIPAL MATHEMATICAL LE0TUBEB
OF 8T JOHN'S COLLEGE, CAMBRIDGE.
THIRD EDITION, REVISED.
GDambrfoge:
MAOMILLAN AND CO.
AND 23, HENRIETTA STREET, COVENT GARDEN,
% H ft U,
1862.
IThe Right of Translation it rwcrwd.^
A TREATISE ON
J>LAHE (^UBIHMTE ifE-@ME-IB¥
AS APPLIED TO THE STRAIGHT LINE
AVD TH1
MEW SI€TI@»S.
Hug ^mntimis (Jranpus.
Bt i T@»minnrsit, M.A.
over jomrs oolubox, Cambridge.
m/JCD EDITION, REVISED.
dambtf 6gt :
MACMILLAN AND CO.
AND 23, MKNEIETT A STREET, COVENT GARDEN,
lonttoit.
1862.
•*
(Kambtftge :
FBINTBD BT 0. J. CLAY, M.A.
AT THIS UNTVEB8ITY PRESS.
PKEFACE.
I have endeavoured in the following Treatise to exhibit the
subject in a simple manner for the benefit of beginners, and
at the same time to include in one volume all that students
usually require. In addition, therefore, to the propositions
which have always appeared in such treatises, I have intro-
duced the methods of abridge*} notation, which are of more
recent origin ; these methods which are of a less elementary
character than the rest of the work, are placed in separate
chapters, and may be omitted by the student at first.
The examples at the end of each chapter, will, it is hoped,
furnish sufficient exercise on the principles of the subject,
as they have been carefully selected with the view of illus-
trating the most important points, and have been tested by
repeated experience with pupils. At the end of the volume
will be found the results of the examples, together with hints
for the solution of some which appear difficult.
The properties of the parabola, ellipse, and hyperbola, have
been separately considered before the discussion of the general
equation of the second degree, from the belief that the subject
is thus presented in its most accessible form to students in
the early stages of their progress.
I. TODHUNTER.
St John's College,
July, 185$.
VI PREFACE.
In the second edition the work has been revised and so
additions have been made both to the text and to the exa
pies; the hints for the solution of the examples have also b(
considerably increased.
March,, 1858.
■
In the third edition some articles which experience proi
to be difficult for students have been simplified and improv
I J and a few additional illustrations have been introduced.
consequence of the demand for the work proving much grea
than had been originally anticipated, a large number of cop
has been printed, and a considerable reduction effected in 1
price.
- ' January, 1864.
•
CONTENTS.
PLANE CO-ORDINATE GEOMETRY.
Chap. Pagb
I. Co-ordinates of a Point - 1
II. On the Straight Line 11
III. Problems on the Straight Lino 25
IV. Straight line continned 58
V. Transformation of Co-ordinates 76
VI. The Circle 84
VII. Radical Axis, Pole and Polar 104
VIIL The Parabola 112
IX The Elapse 141
X. The Ellipse continued. 168
XI. The Hyperbola 189
XII. The Hyperbola continned 203
XIII. General Equation of the Second Degree . 227
XIV. Miscellaneous Propositions 247
XV. Abridged Notation 274
XVI. Sections of a Cone. Anharmonic Ratio and Harmonic
Pencil 300
ANSWERS TO EXAMPLES .... 310
Students reading this work for the first time may omit Chapters
IV, VII, XIV, XV, XVL
* •
PLANE CO-ORDINATE GEOMETRY.
CHAPTER I.
CO-ORDINATES OF A POINT.
R
1. In Plane Co-ordinate Geometry we investigate the
properties of straight lines and curves lying in one plane
by means of co-ordinates ; we commence by explaining what
we mean by the co-ordinates of a point.
/ -V - - ^^v 7
Let be a fixed point in a plane through which the lines
X' OX, Y' F, are drawn at right angles. Let P be any
other point in the plane; draw PM parallel to OF meeting
OX in M y and PN parallel to OX meeting OY in N. The
position of P is evidently known if OM and OAT are known ;
for if through N and M lines be drawn parallel to OX and
OF respectively, they will intersect in P.
The point is called the origin; the lines OX and OF
are callea axes; OM is called the abscissa of the point P; and.
T. C. B. 1
2 (IO-0BDINATES OP A POINT.
ON", or its equal MP, is called the ordinate of P. Also OM
and Jt£P are together called co-ordinates of P.
2. Let OM= a, and O.W= 5, then according to our defi-
nitions we may say that the point P has its abscissa equal to a.
and its ordinate equal to b; or, more briefly, the co-ordinates of
the point P are a and b. We shall often 9peak of the point
which has a for its abscissa and b for its ordinate, as the
point (a, b).
3. A distance measured along the axis OX is however
most frequently denoted by the symbol x, and a distance
measured along the axis OY by the symbol y. Hence OX
is called the axis of #, and OY the axis of?/. Thus x and y
are symbols to which we may ascribe different numerical
values corresponding to the different points we consider, and
we may express the. statement that the co-ordinates of P are
a and b, thus; for the point P, x=a &aA.y = b.
4. The lines X' OX, Y' OY, being indefinitely produced
divide the plane in which they lie into four compartments.
It becomes therefore necessary to distinguish points in one
compartment from points in the others. For this purpose
the following convention is adopted, .which the reader has
already seen in works on Trigonometry ; lines measured along
OX are considered positive and along OX' negative; lines
measured along OF are considered positive, and along OY'
negative. (See Trigonometry , Chap. IV.) If then we pro-
duce PN to a point Q such that NQ — NP, we have for the
point Q, x= — a, y = b. If we produce PM to R so that
MR = MP, we have for the point R, x = a, y = — b. Finally
if we produce PO to 8 so that 0S= OP, we have for the
point 8, x = — a, y = ~b.
5. In the figure in Art. 1 we have taken the angle YOX
a right angle; the axes are then called ■rectangular. If the
angle FOA'be not a right angle, the axes arc called oblique.
All that has been hitherto said applies whether the axes are
rectangular or obliqne. We shall always suppose the axes
rectangular unless the contrary he stated ; this remark applies
both to our investigations and to the examples which are given
for the exercise of the student.
POLAR CO-ORDINATES OP A POINT.
6. Another method of determining the position of a point
in a plane IB by means of polar co-ordinates.
Let be a fixed point, and OX a fixed line. Let P be
any other point; join OP; then the position of P is determined
if we know the angle XOP and the distance OP. The angle
is usually denoted by 6 and the distance by r.
O is called the pole, OX the initial line; OP the radius
vector of the point P, and POX the vectorial angle.
7. The position of any point might be expressed by
positive values of the polar co-ordinates 8 and r, since there
is here no ambiguity corresponding to that arising from the
four compartments of the figure in Art. 4. It is however
found convenient to use a similar convention to that in
Art. 4 ; angles measured in one direction from OX are con-
sidered positive and in the other negative. Thus if in the
figure XOP be a positive angle, XOQ will be a negative
angle; if the angle XOQ be a quarter of a right angle, we
may say that for XOQ, <? = --. It is, as we have stated,
not absolutely necessary to introduce negative angles, but con-
venient; the position of the line OQ, for instance, might be
determined by measuring from OX in the positive direction
Ian angle =27T— g- as well as by measuring an angle in the
negative direction = - .
Also positive and negative values of the radiu9 vector are
admitted. Thus, suppose the co-ordinatea of P to be - and a
4 CO-ORDINATES OF A TOINT.
that is, htXOP=j and OP=a; produce PO to P', so
that OP' = OP, then P' may he determined by saying its
co-ordinates arc — and — a. Thus when the radius vector is a
4
negative quantity, we measure it on the same line as if it had
been a positive quantity but in the opposite direction from O.
Hence if /3 represent any angle and c any length the same
point is determined by the polar co-ordinates /3 and — c aa by
the polar co-ordinates it -f p and c.
8. Let x, y denote the co-ordinates of P referred to OX
as the axis of x, and a line through perpendicular to OX as
the axis of y. Also let $ and r be the p^kr co-ordinates of P.
If we draw from P a perpendicular on OX, we see that
x = r cos 9, and y = r sin 0.
These equations connect the rectangular and polar co-ordi-
nates of a point. From them, or from the figure, we may
deduce
x t +y , = r 3 , ^ = tan 9.
9. We proceed to investigate expressions for some geome-
trical quantities in terms of co-ordinates.
To find an expression for th* length of the line Joining two
points.
axe
<r~7
Let P and Q be the two points; w the inclination of tht
axes OX, Y. Draw PM, QN parallel to Y; let «, , y x be
LENGTH OF A LINE. 8
the co-ordinates of P, and #., y a , those of Q. Draw Pfi
parallel to OX. Then, by Trigonometry,
P^ = Pi?+^-2PB. QBcoaPBQ
= PJ?+ QIP + 2PB. QRcoao.
But PB = a? a — a? t , and (?i2 =y a — y x ; therefore 1 ^ •.
and thus the distance PQ is determined.
If the axes are rectangular, we have
P#-fe-aO» +(&-&)•.' (2).
The student should draw figures placing P and Q in the
different compartments and in different positions ; the equa-
tions (1) and (2) will be found universally true.
From the equation (2) we have
PQ'=x* + y* + x i * + y?-2 (xfr + Mj .'..... (3).
The following particular cases may be noted. ♦
If P be at the origin x x = and y^O; thus
If Pbe on the axis of a? and Q on the axis of y, y x — and
a? t = 0; thus
Let 4 , r x be the polar co-ordinates of P, and a , r a those
of Q\ then, by Art. 8, *
x x = r x cos t9 y x = r x sin l9
a? a = r a cos0 a , y a = r a sin0 a .
Substitute these values in (3) and we have
PQ' = r 1 * + r?-2r 1 r 2 coa{0 2 -0 l ).
This result can also be obtained immediately from the tri-
angle POQ formed by drawing lines from P and Q to the
origin. M.
n
B CO-ORDINATES OP A POINT.
10. To find the co-ordinates of the point which divides in
a given ratio the line joining two given points.
Let A and B be the given points, x,, y x the co-ordinates
of A, and x t , y t tho3e of B; and let the required ratio be
that of n, to n t . Suppose C the required point, so that
AC : CB :: n, : m„. Draw the ordinatea AL, BM, CN; and
AR parallel to OX meeting CN in D. Let x, y be the co-
ordinates of O.
It is obvious from the figure that
LN
AG
: CB>
that is,
Similarly, y = ^^^?-.
n i + n ,
In this article the axes may be oblique or rectangular. A
simple case is that in which we require the co-ordinates of the
point midway between two given points ; then h, = w, and
y=i(y l +y 1 )-
AREA OP A TRIANGLE.
11. To express the area of a triangle in terms of the c
ordinates of its angular points.
Let ABC be a triangle; let a,, y % be the co-ordinates of
A ; x 3 , y, those of B; x s , y a those of C. Draw the ordinates
AL, BM, CN. The area of the triangle is equal to the
trapezium ABML + trapezium BCNM—tYB.pez'mm ACNL.
Y
a
I
Q L M
The area of the trapezium ABML is \LM{AL+BM).
This is obvious, because if we join BL we divide the trape-
zium into two triangles, one having AL for its base and the
other BM, and each having LM i'ur its height;
thus, trapezium ABML = \{x 3 — a,) (y l + y t ) ;
also, trapezium BCNM= £ (a; s — x t ) (y a + tf t ) ;
and, trapezium A ONL »$(%•? a?,) (y, + &) ;
therefore triangle ABO
= i 1K-^,) (y,+?«) + (^-^) (y.+ri - (^ -^i) (y, +?.)]■
Thi3 expression may be written more symmetrically thus ;
i((^-»J(3'«*yJ+(^-«^£Si+yJ+(« 1 -aO(y,+y,)}-( 1 )/
By reducing it, we shall find the area of the triangle
=iky,-v i + a: 1 y.-^ 1 +^.- :c ! y 1 ] ( 2 )-
If the axes be oblique and inclined at an angle a>, the area
of the trapezium ABML = ^LM(AL + BM) sin to, and
laxly for the other trapeziums. Thua ttifc Kreft <A "Cv«.
LOCUS OF AN EQUATION.
i given above
■will be found by multiplying the expressions given
by sin to.
However the relative situations of A, B, C may be changed,
the student will always find for the area of the triangle the
expression (2), or that expression with the sign of every term
changed. Hence we conclude, that we shall always obtain
the area of the triangle by calculating the value of the expres-
sion (2), and changing the sign of the result if it should prove
negative.
Locus of an equation. Equation to a curve.
12. Suppose an equation to be given between two unknown
quantities, tor example, y — x - 2 = 0. We see that this
equation has an indefinite number of solutions, for we may
assign to x any value we please, and from the equation deter-
mine the corresponding value of y. Thus corresponding to
the value3 1, 2, 3, ... of a;, we have the values 3, 4, 5, ... of y.
Now suppose a line, straight or curved, such that it passes
through every point determined by giving to x and y values
that satisfy the equation y— x — 2 = 0; such a line is called
the locus of the equation. It will be shewn in the next
chapter that the locus of the equation in question is a straight
line. We shall see as we proceed that generally every equa-
tion between the quantities x and y lias a corresponding locus.
But instead ot starting with an equation and investigating
what locus it represents, we may give a geometrical definition
of a curve and deduce from that definition an appropriate
equation ; this will likewise appear as we proceed; we shall
talte successively different curves, define them, deduce their
equations, and then investigate the properties of these curves
by means of their equations. We shall in the next chapter
begin with the equation to a straight line.
The connexion between a locus and an equation is the
fundamental idea of the subject and must therefore be carefully
considered; we shall place here a forma] definition which we
shall illustrate in the next chapter by applying it to a straight
line.
Def. The equation which expresses the invariable rela-
tion which exists between the co-ordinates of every point of
EQUATION TO A CURVE.
a curve is called the equation to the curve ; and the curve, the
co-ordinates of every point of which satisfy a given equation,
is called the locus of that equation.
13. The student has probably already become familiar
with the division of algebraical equations into equations of
the first, second, third ... degree. When we speak of an
equation of the n"* degree between two variables we mean
that every term is of the form Ax"yt where a and £ are zero
or positive integers such that a. + B is equal to n for one or
more of the terms but not greater than n for any term, and A
is a constant numerical quantity ; and the equation is formed
by connecting a series of such terms by the signs + and — ,
and putting the result = 0.
EXAMPLES.
1. Find the polar co-ordinates of the points wb
angular co-ordinates are
(1) x = \,y=\; (2) x=-\, y = 2;
(3) a— l,jr«l| (4) x = -l,y = -l;
and indicate the points in a figure.
2. Find the rectangular co-ordinates of the points whose
polar o
i rect-
a)
(3)
-3;
(2) « = -
(4) » r -
and indicate the points in a figure.
3. The co-ordinates of P are — 1 and 4, and those of Q
3 and 7 ; find the length of PQ.
Find the area of the triangle formed by joining the
t three points in question 1.
5. A is a point on the axis of x and B a point on the
axis of y ; express the co-ordinates of the middle point of
AB in terms of' the abscissa of A and the ordinate of B \ alvsw
lso that the distance of thi3 point from tVe, cm^-m =\ AB.
10 EXAMPLES.
6. Transform equation (2) of Art, 11 so as to give
expression for the area of a triangle in terms of the polar
co-ordinates of its angular points. Also obtain the result
directly from the figure.
7. A and B are two points and is the origin ; express
the area of the triangle A OB in terms of the co-ordinates of
A and B, and also in terms of the polar co-ordinates of A
and B.
8. A, B, C are three points the co-ordinates of which are
expressed as in Art. 11 ; suppose D the middle point of AB;
join CD and divide it in so that CG = 'ZGD; find the
co-ordinates of G.
9. Shew that each of the triangles GAB, GBG, 6 A C,
formed by joining the point G in the preceding question to
the points A, B, 0, is equal in area to one-thnd of the
triangle ABC. See Art. 11.
10. A and B are two points ; the polar co-ordinates of A
are 9 lt r.\ and those of B are 0„ r t . A line is drawn from
the origin bisecting the angle A OB; if C be the point
where this line meets AB shew that the polar co-ordinates
of C are « = H», + 8>nd r = "W^l^,! .
11. Find the value of CD* and AD 3 in question 8 in
terms of the co-ordinates there used ; and shew that
12. Find the value of GA % , GB*, and GO 1 , in question
in terms of the co-ordinates there used ; and shew that
3 {GA' + GB'+ GO*) =AB , + BC>+ CA\
( 11 )
CHAPTEE n.
ON THE STRAIGHT LINE.
14. To find the equation to a straight line.
*LQu<*JZ*A
We shall first suppose the line not parallel to either axis.
Let ABD be a straight line meeting the axis of y in 2?.
Draw a line OE through the origin parallel to ABD. In
ABD take any point P; draw PM parallel to OY, meeting
OX in M and OE in Q.
Suppose OB=*c, and the tangent of EOX—m; and let
x, y be the co-ordinates of P; then
y = PM=PQ+QM
= OB+QM
= c+OJftan QOM
= c + mx.
Hence the required equation is
y = mx •+ c.
OB is called the intercept on the ax\& ot oj \ VI ^a& "W^
12 EQUATION TO A STRAIGHT LINE.
crosses the axis of y on the negative side of 0, c will be
We denote by m. the tangent of the angle QOMor BAO,
that is, the tangent of the angle which that part of the line
which is above the axis' of x makes with the axis of x pro-
duced in the positive direction. Hence if the line through
the origin parallel to the given line falls between OF and OX,
■m is the tangent of an acute angle and is positive ; if between
OF and OX produced to the left, m is the tangent of
obtuse angle and is negative. So long as we consider the
same straight line m and c remain unchangeable, they are
therefore called con-itant quantities or constants. But x and
y may have an indefinite number of values since we may
ascribe to one of them, as x, any value we please, and find
the corresponding value of y from the equation y — mx + c
x and y arc therefore called vtruthh quantities or variables.
If the line pass through the origin, c = 0, and the equation
becomes
y = mx.
15. We have now to consider the cases in which the line
i is parallel to one of the axes.
If the line be parallel to the axis of x, m — 0, and the
equation becomes
y = c.
If the line be parallel to the axis of y, m becomes the
tangent of a right angle and is infinite ; the preceding investi-
gation is then no longer applicable. We shall now give
separate investigations of these two cases.
To investigate, the equation to a line parallel to one of the
r
D
B
— d
X
o
J*
1
X
. EQUATION OF THE FIRST DEGREE. IS
First suppose the line parallel to the axis of x. Let BO
be the line meeting the axis of y in B; suppose 02? = ft.
Since the line is parallel to the axis of x, the ordinate PM
of any point of it is equal to OB. Hence calling y the
ordinate of any point P, we have for the equation to the line
y = *.
Next suppose the line parallel to the axis of y. "Let AD
be the line meeting the axis of x in A ; suppose OA = a.
Since the line is parallel to the axis of y, the abscissa of any
point of it is OA. Hence calling x the abscissa of any point,
we have for the equation to the line
x = a.
16. We have thus proved that any straight line whatsoever
is represented by an equation of the first degree; we shall
now shew that any equation of the first degree with two
variables represents a straight line.
The general equation of the first degree with two variables
is of the form
Ax + By + (7=0 (1),
A, B, being finite or zero.
First suppose B not zero ; divide by B, then from (1)
y=-B~B x ( 2 )-
Now we have seen in Art. 14* that if a line be drawn
meeting the axis of y at a distance — -^ from the origin
and making with the axis of x an angle of which the tangent
is — -g> then (2) will be the equation to this line. Hence (2),
and therefore also (1), represents a straight line.
If A = 0, then by Art. 15 the line represented by (1) is
parallel to the axis of x.
If J?=0, then (1) becomes
Ax+C=0 t
EQUATION IN TEEMS OF THE INTEECEITS.
G
and from Art. 15 we knc
parallel to the axis of y.
Hence the equation Ax +
straight line.
that this equation represents a line
+ G= always represents
17. Equation in terms of the intercepts. The equation to
line may also be expressed in terms of its intercepts on the
i-o axes.
Let A and B be the points where the straight line meets
the axes of x and y respectively. Suppose OA = a, OB = b.
Let P be any point in -the line ; x, y its co-ordinates ; draw
I'M parallel to Y. Then by similar triangles,
PM
OB =
AM
AO ;
18. It will be a useful exercise for the student to draw the
straight lines corves* ponding to some given equations. Thus
suppose the equation 2y + 3x = 7 proposed; since a straight
line is determined wiien two of its points are known, we may
fin
EXAMPLES OF STKAlQIIT LINES.
15
find in any manner we please two points that lie on the line,
and by joining them obtain the line. Suppose then x = 1, it
follows from the equation that y = 2; hence the point which
has its abscissa = 1 and its ordinate = 2 is on the line. Again,
suppose x = 2, then y = \ ; the point which has its abscissa
= 2 and its ordinate =^ is therefore on the line. Join the
two points thus determined and the line so formed, produced
indefinitely both ways, is the locus of the given equation.
The two points that will be most easily determined are
generally those in which the required line cuts the axes.
Suppose x = in the given equation, then y = \, that is, the
line passes through a point on the axis of y at a distance i
from the origin. Again, suppose y = 0, then x = \, that is, the
line passes through a point on the axis of x at a distance
\ from the origin. Join the two points tiius determined, and
the line so formed, produced indefinitely both ways, is the
locus of the given equation. What we have here ascertained
as to the points where the line cuts the axis, may be obtained
immediately from the equation ; for if we write it in the form
■
Zx 2#
y+ j =
and compare it with the equation in Art. 17,
x y
we see that a = I and b = \.
Again, suppose the equation y = x proposed. Since this
equation can be satisfied by supposing x — and y = 0, the
origin is a point of the line which the equation represents ;
therefore we need only determine one other point in it. Sup-
pose x=\, theny = l; here another point is determined and
the line can be drawn. The line may afeo be constructed by
comparing the given equation with the form in Art. 14,
y = mx.
This we know represents a line passing through the origin
I and making with the axis of x an angle of which the tangent
is m. Hence y — x represents a line passing through ''
origin and inclined at an angle of 45° to the axis q\ ».
s
inclined )
I - that
16 EQUATION IS TERMS OF THE PERPENDICCL,
Similarly the equation y = — x represents a line incl
the axis of x at an angle of which the tangent is — 1 ; that
at an angle of 135". Hence thi3 equation represents a 1
through bisecting the angle between OY and OX pn
duced to the left in the figure to Art. 14.
19. The student is recommended to make himself t
roughly acquainted with the previous Articles before proceec
ing with the subject. In Algebra the theory of indetermii
equations does not usually attract much attention, and
student is sometimes perplexed on commencing a subject
which he has to consider one equation between two tmltnow
quantities, which generally has an infinite number of
tious.
Our principal result up to the present point is, that a stra!
line corresponds to an equation of the first degree, and
student must accustom himself to perceive the appropr
line as soon as any equation is presented to him. The ]
can be determined by jiscertaining two points through whic
passes, that is, by finding two points such that the co-ordii
of each satisfy the given equation, and the line being
determined, the co-ordinates of any point of it will sa
the given equation.
v-/— f. 20. Equation toastraightlmeintermsoftheperpr-nrJich
fe».Y^/(J> f rom ^' e wigi^ and the inclination of this perpendicular to
—*- — 'axis.
EQUATION IN TERMS OP THE PERPENDICULAR. 17
Let OQ be the perpendicular from the origin on a
line AB. Take any point P in the line; draw PM per-
pendicular to OA, -JCVperpendicular to OQ, and PR perpen-
dicular to MN. Suppose OQ=p, and the angle QOA = a.
Let x, y be the co-ordinates of P; then
OQ= ON+NQ = ON+PB
= OM cos QOA + PMainPMB
= a?cosa+ysina.
Therefore the equation to the line is
x cos a + jysin a = p.
21. We have .given separate investigations of the different
forms of the equation to a straight line in Articles 14, 17, 20 ;
any one of these forms may however be readily deduced from
either of the others by making use of the relations which exist
between the constant quantities. The quantity which we
have denoted by b in Art. 17, that is QB> is denoted by c in
Art. 14 ;
.-. b = c (1).
In Art. 17, v ..
- = tan BA = tan (tt - BAX) / -y ..•"<-,
/ ' s ■ 4 > ' \. •
= — tonBAX; \*J--'- -
in Art. 14 we have denoted the tangent of BAX by m)<<;>- /
... - = -7W (2).
In Art. 20, GA cos a = OQ, and &Bsina= OQ; that is,
^=5 a cosa = 5 sin a (3) ;
therefore from (2) and (3), m = — cot a (4).
Also if the equation
Ax + By+ (7=0,
T. a & ^
t
18 OBLIQUE CO-OKDINATES.
represent the straight line under consideration, then by
Art. 16,
-^ = m, _-g = C = 5 (5);
A A° P
.". -p=cotaand •= = — /— (6).
B B sin a ^ '
By means of these relations we may shew the agreement
of the equations in Art3. 14, 17, 20, or from one of them
deduce the others.
22. The student may exercise himself by varying the
figures which we have used in investigating the equations.
Thus, for example, in the figure to Art. 17, suppose the point
P to be in BA produced, so that it falls below the axis of x.
We shall still have
PM_ AM PM_ x-a ,
OB~AO' ° r b ~ a •
Now atnce P is below the axis of x, its ordinate y is
a negative quantity, hence we must not put PM = y but
PM= — y, because by PM we mean a certain length esti-
mated positively. Thus
b~
i before,
ique Co-ordinates.
tliit Crrviis. 23. Equation to a straight line.
Vi t"*rXi£i yf e 3na n fo^te the inclination of the axes by w.
Let.
a lin
Suppose first, that the line is not parallel to either axis.
IjntAPP be a straight line meeting the axis of y in B. Draw
a line OE through the origin parallel to ABB. In ABD
EQUATION TO A STRAIGHT LINE.
19
take any point P; draw PM parallel to OY, meeting OX in
M and OEm Q. Suppose 0-8 = c, and the angle QOM—a.
But
Let x, y be the co-ordinates of P; then
y=PM=PQ+ QM= 0B+ QM.
QM sin a
• •
0M~ Bin (•-a)'
n ,- a; sin a
sin(tt> — a)
Hence the required equation is
x sin a
sin (© — a)
sin a
If we put m for — — ; r the equation becomes
r sm (© — a) ^ L
y = mx + c,
as in Art. 14. The meaning of c is the same as before ; m is
the ratio of the sine of the inclination of the line to the axis
of a; to the sine of its inclination to the axis of y. Since
sin a is always positive, m will be positive or negative accord-
ing as sin (a> — a) is positive or negative ; thus as before to,
will be positive or negative according as the line through the
origin parallel to the given line falls between OY and 0X y
or between Y and 0X\ The meaning of m coincides with
IT
that in Art. 14 when a> = — , for then m = tan a.
%—*
20 EQUATION IN TERMS OF THE PERPENDICULAR.
,. a . sin a
24. Since «* = -.— . r-;
sin (w - a)
.". m (sin w cos a — cos <w sin a) = sin a ;
.', m (sin e> — cos w tan a) = tan a ;
m sina>
.*. tan a = — .
" ± V(l + 2m cos <u + m") '
+ V(l + 2i» cos a + m')'
Since sin a is positive, we must take the upper or lower
sign according as m is positive or negative.
25. The investigations in Arts. 15 and 17 apply without
alteration to the case of ohlique axes, and those in Art. 16
with the requisite change in the meaning of the constant m.
26. To find the equation to a straight line in terms cfthr
perpendicular from the origin, and the inclinations of the per-
pendicular to the axes.
Let 0Q he the perpendicular from the origin on a line AS;
let Q = p, OA = a, OB = b. If we suppose Q OA = a, we
have QOB = w — o ; denote this by ft ; then
SYMMETRICAL EQUATION TO A STRAIGHT LINE. 21
00 = a cob a: .% a = -^—.
^ ' cos a
0# = Jcos£; mJ = A.
Substitute in the equation, Art 17,
and we obtain
x cos a + y cos/8 =p.
27. The following form of the equation to a straight line
is often useful.
Let Q be a fixed point in any line AB) h, h its co-ordi-
nates ; let P be any other point in the line ; x, y its co-
ordinates; let QP=r, and the angle BAX—a* Draw the
ordinates PM, QN; and QR parallel to OX; then
x — h sin (to — a) T
-. — ^ l bb j suppose.
r sin© rr J
V — & sin a
* s- -j — -. w suppose,
r sin© rr
thus — i — = * =r«
« n
22 POLAE EQUATION TO A BTRAIGHT LINE.
For the equation to the line it is sufficient to put
tc — h _ y — h
I n~'
hut it is useful to remember that each of these quantities
equal to r.
If the axes are rectangular, I and n become respectively
cos a and sin a, that is, the cosines of the inclinations of the
line to the axes of x and y respectively.
In the preceding figure P tails to the right of Q and x — A
is positive. If P were to the left of Q then x — h would be
negative. Thus since x — h — lr, the product lr must be
capable of changing its sign; this leads us to consider r
positive or negative according to circumstances. "When there-
fore we write the equation to a straight line under the form
I n '
and ascribe to I and n the values given above, we conclude
that each of the expressions — ; — and ~ is numerical
equal to the distance between the point (A, 7c) and the point
(x, y), but that the sign of each expression will depend upon
the relative positions of the two points.
\ t L^<i.itn~
Polar Co-ordinates.
Polar equation to a straight line.
Let AB be a straight line, OQ the perpendicular on it
from the origin, OX the initial line, P any point in the line.
EOEMS OF THE EQUATION TO A STRAIGHT LINE, 23
Suppose OQ=*p> and the angle QOX= a. Let r, be the
polar co-ordinates of P; then
QQ=OPcosPOQ;
that is, y = rcos(fl — a). *
This is the polar equation to the line.
29. The polar equation may also be derived from the
equation referred to rectangular co-ordinates. Let
Ax + By + (7=0
be the equation to a line referred to rectangular co-ordinates.
Put rcosfl for x, and rsinfl for y, Art. 8; thus
^4rcos0 + .Brsin0+ (7 = (1)
is the polar equation. This equation may be shewn to agree
with -
j p = rcos(5 — a) (2).
For by Art. 21 we have
x A . , G p
-^ = cot a and -^ = — r — .
B B sin a
Hence (1) becomes
cotarcos0+rsin0 — J— =0,
sin a
which agrees with (2).
30. The equation to a line passing through the origin
is, by Art. 14,
y=*mx.
Put rcosfl for x and rsinfl for y\ the equation then
becomes
r sin = m r cos $ ;
•\ tan0 = m;
•\ 5 = a constant ;
this is therefore the polar equation to a line passing through
the origin.
jr. fi X + fy+L* O /IU0 XfrS*+),sC^4i-)> ■*&
V
24 EQUATIONS TO STRAIGHT LIKES. EXAMPLES.
31* We will collect here the different forms of the eq
tion to a straight line which have been investigated,
V Y = ma?4-c, Arts. 14 and 23
a? = constant, or, y = constant, Arts. 15 and 25
- + f-l=0, Arts. 17 and 25
a b \
x cos a +y sin a —p = 0, Art. 20.
y = -T—7 r a? + c, Art. 23.
* Bin(a>-q) ]_
¥ g cos q+y c os ft — p = 0, Art. 26.
—^ = 2 = r, Art. 27.
p = rcos(0 — q) , Art. 28.
. 4r cos fl + .Br sin 4- = 0. Art. 29.
= constant, Art. 30.
EXAMPLES.
Draw the straight lines represented by the followi
equations :
(1) y + 2a; = 4; (2) 2y-a; = 2;
(3) y + a?=-2; (4) »-2y = 4;
(5) y+2a; = 0; (6) l = cos^-~);
(7) »-l; (8) = J;
(9) = 0. (10) 0=1.
CHAPTER in.
PROBLEMS ON THE STRAIGHT LINE,
"We proceed to apply the results of the preceding
articles to the solution of some problems.
To find the form of the equation to a straight line which ' '
passes through a given point . ' ^"T^L*
Let x 1 , y x be the co-ordinates of the given point, and
suppose
y = mzc + c (1)
to represent the straight line. Since the point (a;,, y,) is 01
the line, its co-ordinates must satisfy (1) ; hence
y^mx^+c (2).
By subtraction,
,-V.=m(x-z.) (3);
this is the required equation.
33. The equation (3) of the preceding article obviously
represents what is required, namely, a line passing through
the point (a:,, y,). For the equation is of the first degree in
the variables x, y, and therefore, by Art. 16, must represent
some straight line. Also the equation is obviously satisfied
by the values * = *,, y = y l ; that is, the line which the
equation represents does pass through the given point. The
constant m is the tangent of the angle which the line makes
with the axis of x, and by giving a suitable value to m we
may make the equation (3) represent any straight line which
passes through the assigned point.
26
EQUATION *0 A STRAIGHT LINE
The geometrical meaning of equation (3) is obvious. For
let AB be any straight line passing through the given point
Q. Let P be any point in the line ; x, y its co-ordinates.
Draw the ordinates PM, QN; and QB parallel to 0X\ then
PR
^g = tangent PQB;
that is, y -^y± _ tan BAX= m,
x — x t
which agrees with equation (3).
34. In Art. 32 we eliminated 6 between the equations (1)
and (2) and retained m; we may if we please eliminate m
and retain c. From (2)
m =
x.
Substitute in (1), thus
x.
.*. yx x -xy x + c{x-x x )-0.
This equation obviously represents a straight line passing
through the given point, because it is an equation of tne first
degree and is satisfied by the values x = x x , y = y x *
35. To find the equation to the straight line which pauses
through two given points .
WHICH PASSES THEOUGH TWO GIVEN POINTS. 27
Let x x , y x be the co-ordinates of one given point; x 2 , y z
those of the other; suppose the equation to the straight line
to be
y = mx+c (1).
Since the line passes through (x x , y x ) and (a? a , y a ),
y x = mx x + c (2),
y a = #w? a + c (3).
From (1) and (2) by subtraction,
y~y t = m{x-x x ) (4).
From (2) and (3) by subtraction,
hence m = — — 2i»
Substitute the value of m in (4) and we have for the
required equation
y-yi-fhdh*-^ ( 5 )-
'a ""l
We may also write the equation thus,
fo-*i)(y-yi) = (y a --yi) (*-*i) («)•
Sonle particular cases may here be noted* Suppose y,=y 1 ,
then (6) becomes (x 2 —x x ) (y—y 1 ) = 0, therefore v = y 1 ; the
required line is thus parallel to the axis of x. Similarly if we
suppose x a =x x , then (6) becomes (y a — y) (a? — x x ) = } there-
fore x = x x ; thus the required line is parallel to the axis of y.
Lastly, suppose the point (x x ,y x ) to be the origin; hence
a? =0 and y x =0; thus (6) becomes x y=y a x. The student
should illustrate these particular cases by figures.
36. The equation (6) of Art. 35 becomes by reduction
x xV - a^i + x &i - *$% + a^ a - x & *= °«
If we compare the expression on the left-hand side of this
equation with the expression in brackets in equation (2\ of
PARALLEL LINES.
Art. 11, we see the only difference is that we have x and y in
the place of x s and y t respectively. Thus the equation
informs us that the area of the triangle formed by joinkj
[x, y), (x lt y,), (x,, y t ) vanishes, as should evidently be tb
case since the vertex (3;, y) falls on the base, that is, on tin
line joining (x lt y t ) to {x t , yj.
37, To find the equation to the straight line which j
through a given point and divides the line joining two other
given points in a given ratio.
Let (h, k) be the first given point ; let (x, , y ) , (x a , y t ) be
the two other given points ; let the given ratio in which the
line joining the last two points is to be divided be that of n
to n % ; then, by Art. 10, the co-ordinates of the point o
division are
A "'y« +w »yi
n, + «
Hence by equation (5) of Art. 35 the equation required is
. .. "iy.+ w «y j.
y-k= " l | W l (x-h);
* n l x 1 + n % x 1 _ ,
* «, (x t — h) +n t (x l - kj * '
To find, the form of the equation to a straight lint
which is parallel to a given straight line.
Let the equation to the given straight line be
y = m 1 x + c l (l),
and the equation to the other straight line
S~mx + e (2).
Since the lines represented by (1) and (2) arc parallel, they
moat have the same inclination to the axis of x; hence
INTERSECTION OF STRAIGHT LINES.
Thus (2) becomes
y as m x x + c.
The quantity c remains undetermined since an indefinite
number of straight lines can be drawn parallel to a given
straight line.
39. To determine the co-ordinates of the point of intersec-
tion of two given straight line s.
Let the equation to one line be
y = m 1 x + c l ...(1),
and the equation to the other
y = ma aj+c 2 (2).
The co-ordinates of the point where the lines intersect
must satisfy loth equations ; we must therefore find the values
of x and y from (1) and (2). Thus
c t — c 2 cfto- - em
these are the required co-ordinates.
40. To find the condition in order that three straight lines
may meet in a point.
Let the equations to the lines be respectively
y = m 1 x + c l . (1), y = w a » + c 2 (2),
y = mjc + c z (3).
The co-ordinates of the point of intersection of (1) and
(2) are
'• m % — m x * m 2 —m l
If the third line passes through the intersection of the first
and second, these values must satisfy (3). Hence the neces-
sary and sufficient condition is
— *- -j- c a ,
30
that is,
ANGLE BETWEEN GIVEN LINES,
c x m % — cjrn 1 + cjm z — c z m % -f c z m x — c^n z = 0,
4- 41, To find the angle between two given straight lines*
Let ABC be one line and DEC the other; let the equat
to the former be
and the equation to the latter
y = m % x + c % .
Then tmAGD = tan ( G4X- CDX)
tan CAX- ten GDX
~~ 1 + tan 04Xtan <72?JC
1 + ffljftlg
From this we may deduce
eos^CZ> =
sin ACD =
1 + W^m,
V{(i+0(i+<)}'
V{(i + 0(i+0}'
42. 2b ^wrf $fo /orw of the equation to a straight li\
which is perpendicular to a given straight line.
IJNE PERPENDICULAR TO A .GIVEN LINE;
81
Let y = mx + c
Tie the equation to the given line, and
y=m'x + c'
the equation to another line. Then the tangent of the angle
"between these lines is
iu — fit
1 + mm' "
If these lines are perpendicular,
Hence
• . m = •
x ,
y = — + c
represents a line perpendicular to the line
y =* mx + c.
43. The result of the last article may also be obtained
thus,
[Y
Let AB be the given line, so that tan BAX= m. Let CD
be a line perpendicular to AB; then
tan D CX=- tan D GO
= - cot BAO
m
32
I
LINE PERPENDICULAR TO A GIVEN LINE*
Hence the equation to CD is
x
where
m
c' - 02>.
.\
44. 21) ,/JnJ £fo equation to the straight line which paas
through a given point, and is perpendicular to a given straiqk
line.
Let x t , y l he the co-ordinates of the given point, aiid
y = mx + c (1)
the equation to the given line. The form of the equation ti
a line through (x l9 y x ) is
If (2) is perpendicular to (1), we have
m'm +1 = 0.
Hence the required equation is
1 f N
(2).
45. To find the equations to the straight lines which vat
through a given point and make a given angle with a a iva
straight line±
EQUATIONS TO CERTAIN LINES.
S3
Let AB be the given straight line ; the given point ;
-,iit8 co-ordinates; ft the given angle. Let the equation
oAB\n
y = mx + c.
Suppose CD and CE the two lines which can be drawn
urougn G, each making an angle ft with AB. Then
tan CDX= tan {BAX±g)
m + tanft
1— m tan^S*
tan GEX=-tan 0J?4--tan(£-.B4X)
_ we — tan ft
""l+mtanft*
Hence the equation to CZ> is
7 m + tanft , TN
^-^ = l- TO tan/3 ^--^'
e equation to CE is
7 f/i •— tan ft
46. The following particular cases of the preceding results
lay be noted.
(1) Suppose m = 0; then the given line is parallel to the
sis of x. The required equations then are
j-i = tan ft (x — .A),
and y — & = — tan ft (#_-lA).
(2) Suppose m = oo ; then the given line is parallel to the
sis of y. And since
-j.t».A l + -tanft
m + tan p m
1 — m tan ft 1 ~
tanft
in
T. C.8.
34 EQUATIONS TO CERTAIN LINES.
we have when m = oo , and therefore — = 0, for the eq nation
to CD,
Similarly the equation to CE becomes
y- &=*cotft(x — h).
(3) Suppose m = tan ft. In this case the equation to CD
becomes
that is, y — A; = tan 2)8 (a? — A).
The equation to CE becomes
y-& = 0,
so that CE is parallel to the axis of x.
(4) Suppose m = cot ft. The equation to CZ> may lie
written in the form
{y — k) (1 — m tan ft) = (m-f tan ft) (x — A),
and we see that when 7w = cotft the left-hand side is zero;
thus the required equation is then
x — h = 0.
The equation to CE becomes
, cot ft— tan ft, 7X
y-*= 2 ^""^
_ cos 8 ft — sin 2 ft ,
~ 2cosftsinft *""*'
= cot 2ft (a* -A).
(5) Suppose m = — tan ft. Then the equation to CD
becomes
y-fc = 0,
EQUATIONS TO CERTAIN LINES. 35
and the equation to CE becomes
= -tan2#(a?-A).
(6) Suppose m = — cotft Then the equation to CD
becomes
, tan j8 — cot 5 , , N
y-* = ^ £(*-*)
= -cot 2fi(x-h).
The equation to CE may be written in the form
(y— &)(X+7»tan£) = (m — tan£) (a? — A),
and we see that when *» = — cot£ the left-hand member is
zero ; thus the required equation is then
x — h — 0.
(7) Suppose fi = — . The equation to CD may be written
, w cot fl + 1 , , x
y - * = — T^ (a - A) .
* cotp — m v '
When £ = — we have cot £ = 0; thus the equation becomes
Similarly the equation to CE takes the same form ; and thus
the result agrees with that of Art. 44.
We have discussed these particular cases as an example of
the manner in which the student should test his comprehen-
sion of the subject by applying the general formulae to special
examples. He will find it useful to illustrate these cases by
figures,
3—2
V.
36
LENGTH OF A PERPENDICULAR
47. To find the length of the perpendicular
Iven point upon a fliven straight line .
drawn Jroma
gtven
/
r
:>
•
R
o
M
X
Let AB be the given straight line ; D the given point;
h, h its co-ordinates. Let the equation to AB be /
y = mx + c
a).
The equation to the line through D perpendicular to Jt
is, by Art. 44,
(2).
\
Let x lt y % be the co-ordinates of E; then, by Art. 9,
DW- {x t - *)•+&-*)» (3).
It remains then to substitute for x. and y x their values in
(3)» Now, since x l9 y. are the co-ordinates of JS 9 which is
the point where (1) ana (2) meet, we have
y^mxt + c, andy l -i = --(a? 1 -i);
•\ mx.+c^k fo — h):
•\ a?.
iwA?H- A — mc
1+m* 9
W
LENGTH OF A PERPENDICULAR. 37
mh — m*h — mo
andajj — A =
1 + m 1
m
1 + m
-, (A?— i»A — c).
Also
m*& 4- wiA + e
fc-Mk + o r+^i— »
and v t — k = — — — =— :
••• * » ""- (ITS? * — *-* ffi jjS=j£
_ (£— wA — c) 2
-" 1+m* *
Hence DE= ~ , T .
V(l + »0
The radical in the denominator may he taken with the posi-
tive or negative sign, according as the numerator is positive
or negative, so as to give for I)E a positive value.
We may also obtain the value of DE thus ; draw the ordi-
nate DM meeting the line AB in H; then
DE=DH*mDHE=DHQMHAM*
Now OM=h; .: HM = mh + c, and DM — It;
.'. Z?2Z*=5 h — mh — c.
Also tan HAM— m : .\ cos HAM = -77- JT ;
V (1 + m)
Hence if on the line y — mx — c = a perpendicular be
drawn from the point (h t ,kj and also a perpendicular from the
point (A 2 , kj, the ratio of the first perpendicular to the second
is equal to the numerical ratio of Jc t — m\ — c to h % — m\ — c.
LENGTH OF A CERTAIN LINE.
48. To find the length of the line drawn from a given point
in a given direction to meet a given line.
Let (h, k) be the given pokt ; and suppose a line drawn
from this point at an inclination a to the axis of cc to meet
the line
Ax + By+C=0 (l).
Let r be the required length; x i ,y i the co-ordinates of
the point where the line drawn from {h, k) meets (I) ; then.
by Art. 27,
a l — A = rcos a, y % — k = r sin a (2).
But (x lt #,) is on (I),
.-. A {h + r cos a) + B (k + rsina) + 0=0; r
Ak+Bk+C
Aco&a + B&ina'
49. In this chapter we have used equations of the form
y = mx+o to represent straight lines. The student maj
exercise himself by solving the problems by means of the
more symmetrical forms of the equation to a straight line,
Ax + By+G=*Q,
%f-l=0,
a o
x cos a + y sin a — p = 0.
The results of course can be easily compared with those we
have obtained. For example, if in Art. 47 we represent the
given line by the equation
Ax + By+O=0,
the result obtained should coincide with the value of
, /.: — mh — c
wheu for m we write
must ba
that is, the result
RULE. FOR TRANSFORMING AN EQUATION. 39
Similarly, if the given line be represented by
x cos a + y sin a — p = 0,
we shall find for the perpendicular on it from (A, h)
± (h cos a + h sin a — p).
Thus if the equation to a line be in the form
x cos a 4- y sin a — jp = 0,
the length of the perpendicular drawn from a point on this
line is the numerical value of the expression on the left-hand
side of this equation, when for x and y are substituted the co-
ordinates of the given point This result is of such great im-
portance that we shall proceed to examine it more closely.
50. We may however previously observe that if the equa- ~~^~
tion to a line be given in any form, we can immediately trans-
form it so that it may be expressed in terms of the length of
the perpendicular from the origin and the inclination of this
perpendicular to the axis of x. For example, suppose the
equation to be
2a? + 3y+4 = 0.
Change the sign of every term so that the last term may be
negative; thus the equation becomes r/l*+2!fy*£-d ft
-2x-3y-4 = 0. *W>^/xJ^<z c/ fit) £
Divide by V(2* + 3*) ; thus i *£*1 f -/ **
V13 V13 V13 * A_c*** <&* s JzJ>
This is of the form jf«t«*~ + -^V^
ojcosa + ysina- - p = 0, ±-J~~~z r « ^ <•
2 . 3
and cos a = — rrr , sin a = —
V13' V13
4 ' ' \ C*3+ * -4—
V13 (#C*-**2^
3*7T
In this example a is an angle lying between ir and — .
' A '-tn
40
LENGTH OF A PERPENDICULAR.
Any other example may be treated in a similar manner- 1
the rule being the following. Collect the terms on one side, I
and if necessary, change the signs ao that the equation may I
be in the form Ax + By — O = 0, where is positive; then!
divide by >J(A* + B*); thus the equation becomes
Ax
this is of the required form, and
A . B
~W)
= 0;
</{A* + B*)'
*/{A* + B*)'
v<^+i<r
Thus every equation representing a straight line may be I
brought to the form
x cos a + y sin a — p = 0,
where p is a positive quantity, unless the line, passes through |
the origin, and then p = 0.
W/ien we use the equation
a; cos a + ysinct— J> =
we shall always suppose p positive.
51. The line whose equation is
x cos a. + y sin a —p =
might be called "the line (p, a)," since the constants p and a
determine the line; but when there. is no risk of confounding
it with another line, it may be more shortly called " the line
a," and the equation may be expressed shortly, thus, " a = 0."
We shall now give another investigation of the expression
for the perpendicular from a given point on the line {p, a).
Let AB represent the line (p, a), the origin, P the
Eoint (x, y), so that P and are on opposite side3 of AB.
•raw OQ, PZ perpendicular to AB, and PM parallel to 01';
through M draw a line parallel to AB, meeting OQ and PZ,
produced if necessary, in Q' and Z' respectively.
LENGTH OF A PERPENDICULAE.
41
Then OQ' = 0Jfcosa=a?cosa; PZ'»PJfsina=ysina;
PZ=0#+PZ'-0# = *cosa + ysina--p.
If P and be on the same side of AB we shall obtain for
he perpendicular
p — x cos a — y sin a.
It will be found that these results will hold for all varieties
f the figure.
52. Or we may proceed as follows.
Let a cos a + y sin a— p = (1)
e the equation to a straight line, and let x, y' be the co-
rdinates of the point from which a perpendicular is drawn
pon the line ; it is required to find the length of this per-
endicular. The equation to any line which is parallel to (1)
nd on the same side of the origin, may be written thus,
a: cos a 4- y sin a— p f = (2),
r here p is the perpendicular from the origin upon this line,
f this line pass through the point (x\ y')> we must have
x' cos a +y' sina — p' = ;
.\ y = aj'cosa + y'sina.
The length of the perpendicular from i^x\ tf) on (1) will be
' — p if the- point and the origin are on different sides of the
ne, and j? —p if they are on the same side ; that is,
x' cos a + y r sin a —p
i the former case, and in the latter case
p — x' cos a — y' sin a.
42 LENGTH OF A PEIiPF.Nnit'UI.AR.
If the line parallel to (1) be on the opposite :
origin, its equation will be
x cos (tt + a) + y sin (tt + a) —p' =
where p' is the length of the perpendicular from the o
upon it. If this line pass through the point (a:', y') i
must have
x' COS a -f y' sin a + p' = ;
.*. p' = — x' cos a — y' sin a.
The length of the perpendicular from {x,y') on (I) will In
the sum of p and p, that is,
We may now suppress the accents on :
have the same conclusion as before.
and y, and in
53. Thus the perpendicular from the point (a;, y) on tb
line
x cos a + y sin a— p =
is x cos a + y sin a — p, or p — x cos a — y sin a,
according as the point (x, y) and the origin are on differs
sides of the line or on the same side of it.
The student will perceive that we speak here of the po
(a:, y) and the line x cos a + y sin a — p = 0, and that we o
the same symbols x, y, in speaking of the point and of t!
line. But we do not mean that the point (x, y) is to be
the line, that is, we do not mean the x and y which are <
ordinates of the point (x, y) to have the same values as th
have for any point in the line a: cos a + jsina — p =0. ^
might use x, y as co-ordinates of the point to prevent co
fusion, but it is found convenient to adopt the notation hi
used, as the advantages more than compensate for the i
creased attention which is required from the student in d
tiuguishing the different meanings of the symbols.
54. "We have in Art. 51 left the student to convince hli
self by drawing the figures in different ways, that the p
pendicnlar from the point (a:, y) on the line (p, a) is alwayi
+ (xcosa,+y sin a -
OBLIQUE AXES.
43
"the upper or lower sign being taken according as (x, y) and
-the origin are on different sides, or on the same side of the
line (p, a). We may also arrive at the result imperfectly,
■thus. AVe may first prove, as in Art. 47, that the perpendi-
cular must always be equal to one of the two expressions
± (x cos a + y sin a — p),
and may then proceed to distinguish the cases. Now the
expression x cos a +y sin a— p is negative when the point
{x, y) is the origin, because it becomes then— p; also this
expression cannot change its sign so long as (x, y) is taken on
■the Bame side of the line (p, a) as the origin because it can-
not change its sign without -patsing through the value zero,
and it cannot vanish until the point [x, y) is on the line.
Hence the expression remains negative so long as (x, y) is on
the same side of the line (;», a) as the origin. Similarly, if
the expression is positive when the point (x, y) has any one
position on the other side of the line [p, a), it will continue,
positive so long as (x, y) is on that side of the line ; and it
may he easily shewn that the expression can be made posi-
tive by suitable values of x and y ; hence it is always positive
while (x, y) is on the opposite side from the origin. We call
this an imperfect method, because the sentence in italics on
which the method depends, has probably not sufficiently at-
tracted the student's attention up to this period of his studies
to produce perfect conviction.
55. If the equation to a line be x cos a +y sin a = 0, so
that p = 0, we shall still have +■ (a? cos a + y sin a) as the
length of the perpendicular from the point (x, y) on it. We
may discriminate as follows, let the equation be so written
that the coefficient of y is positive ; then for points on the
same side of the line as the positive part of the axis of y, the
perpendicular is x cos a +y sin a ; for points on the other side
it is — (a: cos a + y sin a) . This is easily shewn by comparing
a few figures, or as in Art. 54.
Oblique Axes.
56. The results in Arts. 32 — 40 hold whether the axes are
rectangular or oblique; in Art. 33, however, m must have
that meaning which is required when the axes are oblique.
44
OBLIQUE Axis.
To find the angle between two straight limes rtfiarrdl
oblique axes.
Let a> be the angle between the axes; y^m 1 x-\-c x l k
equation to one line; y = mfc + c 9 the equation to the otto,
Let otj, a. be the angles which these lines make with the nil
of x ; and fi the angle between them.
By Art. 24
tn, sin o> . Hasina
tana t =; — * : tan ol=— t 3 .
1 1 + WljCOSOI ^ 1 + mucosa*
m t 8ino
Hence tan £ or tan (a, — aj = '
cosa>
at. sin a
1 -f- W t CQ8#
1 +
m l m^ sin 8 a>
(1 + WjCOsa) (1 + m^coiij
(m, — yi t ) sin co
1+ ( wij + «ij) cos a> + m^ '
Hence the condition that the lines may be at right angles is
1 + (w» t + fli a ) cos (o+ m x m^ 0.
57. To find the length of the perpendicular drawn front
given point on a given straight line.
We shall proceed as in the latter part of Art, 47 ; tk
student may also obtain the result by the method in tk
former part of that article*
Let AB be the given straight line ; D the given poinl
h, k its co-ordinates.
"•*«.
a *-,
I
46 POLAE CO-ORDINATES,
that is, reos0cosa + )-sin0Bin<z=^> 1 .
Since this line passes through the two points, we have
r, cob0, cos a + r 1 sin 8 l ama=p (J
r a cos 0, cos a -t-r a sin 0,8^01=^) (3
From (1) and (2)
(r cob 8 — j-[ cos 0J cosa + (r 3in 8 — r l sin t ) sina = 0...(t
From (2) and (3)
(r 1 cos0 1 — r a eos0 1 )cosa-j-(r 1 Bin0 1 — f B sin^Jsirt a = 0...(i
r cos — r t cos 0, r sin 8 — r 1 sin 9 i
r, cob 9 l — r s cos 0, r, sin 0, — r t sin B
After reduction we obtain
w, sin {0 t - 0) + r,r, sin (0 8 - 8,) + r % r sin (0 - 0J = ...((
This equation lias a simple geometrical interpretation ; '
if we draw a figure and take for the origin, and A, B, Pi
the points ()-,, 0,), (r a , a ), (j% 0), respectively, we see tl
equation (6) is the expression of the fact that one of the
angles OAP, OBP, OAB, is equal in area to the sum of ti
other two.
59. We have seen that a straight line is the locus of:
equation of the first degree ; as we proceed it will appear th
if an equation he of a degree higher than the first, the C(
responding locus will be generally some curve; we may noti
here some exceptional cases.
Suppose the equation
a? — lose + 4a 1 -f y* =
be proposed ; this equation may be written
(aj-2o)' + / = 0.
Hence we see that the only solution is
EXCEPTIONAL CASES. 47
Thus the corresponding locus consists only of a single point
q the axis of a; at a distance 2a from the origin.
Again, suppose the equation to be
No real values of x and y will satisfy this equation; in this
ase then there is no corresponding locus, or as it is usually
xpressed, the locus is impossible. Thus, the locus corre-
ponding to a given equation may reduce to a single point, or
; may be impossible.
60. We have seen that the equation to a single straight line -f-
$ always of the first degree ; an equation of a higher degree
han the first may however represent a locus consisting of two
r more straight lines. For example, suppose y
^-^=0 (i);
.'• y = « (2), ory = -a? (3).
If the co-ordinates of a point satisfy either (2) or (3), they
rill satisfy (1) ; that is, every point which is comprised in
he locus (2) is comprised in (1), and every point which is
omprised in (3) is also comprised in (1). Hence (1) represents
wo straight lines which pass through the origin, and make
espectively angles of 45° and 135° with the axis of x.
61. An equation which only involves one of the variables,
©presents a series of lines parallel to one of the axes. Thus,
f there be an equation f{x) = 0, we obtain by solving it a
eries of values for a?, as x — a x or # = «., and each of
hese equations represents a line parallel to the axis of y.
Similarly f(y) = represents a series of lines parallel to the
kxis of x.
An equation of the form f f-J = represents a series of
ines passing through the origin ; for by solving the equation
ve obtain a series of values for 2- , as - = m. , — = m 9i .... and
xx l x 2
sach of these equations represents a line passing through the
J ^ /> : _ xdUtLs ft^^is \**^- ^ '
n V^uAAi
/(!
LOCI CONSISTING OF STEAIQHT LINES.
origin. Of course if an equation f(x) = 0, f{y) = 0, or
have no real roots, the corresponding locus is im-
possible.
The equation
Ay* + Bxy + Ox* = Q
may be put in the form
a(¥)'+b¥ + c=o.
Since this is a quadratic in - we obtain two values for it
suppose — = mj and - = m > ; hence the equation generall;
represents two straight lines passing through the origin.
1? be less than AA C, then m, and m s are impossible, and 1
only solution of the given equation is x = 0, y — ; that
the locus is a single point, namely, the origin.
62. It is obvious that if the locus represented by an equ*
tion/(as, y) = passes through the origin, the values x=l
f = must satisfy the equation. We can thus immediate
etermine by inspection whether a proposed locus passe
through the origin or not.
63. In Art. 39 we determined the co-ordinates of the poin
of intersection of two given straight lines : the proposition
may obviously be generalised. Let_/| (x, y) = 0, j£(ar, y) =0,
represent two curves, then the co-ordinates of the points where
they meet will be determined by solving these simultaneous
equations. It may be shewn that if one equation be of the
m" 1 degree and the other of the n Xh , the number of common
points cannot exceed mn. {See Theory of Equations, Chapter
XX.)
64. We will exemplify the articles of this chapter by
applying them to prove some properties of a trianr 1 "
The lines drawn from the angles of a triangle to tlte middle
points of the opposite sides meet in a point.
Let ABC be a triangle, J), E, F the middle points of the
sides ; take A for the origin, AB for the direction of the
PROPERTIES OF A TRIANGLE.
49
of Xj and a line through A perpendicular to AB for the axis
of y. Let AB=a, and let x, y be the co-ordinates of (7.
Since D is the middle point of GB, the abscissa of D is
J (#' + a) and its ordinate 2- (Art 10) ; since E is the middle
. point of -4 (7, the abscissa of J? is — and its ordinate ^- ; since
* F\& the middle point of AB, its abscissa is - and its ordinate
zero. Hence by Art. 35,
the equation to AD is y = -r — •••«
3/ T fl
the equation to BE is y *= ^4 — — '
the equation to OF is y= ^ f_ q '
■(*);
■(»)•
To find the point of intersection of (2) and (3) we put
y' (g - «) _ y' (2s - <0 .
x' — 2a 2x' - a '
•\ (a?-a) (2a?'-a) = (2a?-a) (a?' — 2a);
•\ 3aa; = a (»' + a) ;
.\ a; = £(a5' + a).
Substitute this value of x in (2) and we find
y'
T. C. S.
50
PROPERTIES OF A TRIANGLE.
We have thus determined the co-ordinates of the points
intersection of (2) and (3) ; moreover we see that these value
satisfy (1); hence the line represented by (l) passes througt
the intersection of the lines represented by (2) and (3), '
proves the proposition.
The lines drawn from the angles of a triangle perpendmh
to the opposite sides meet in a point.
The equation to BC is (Art.,35)
hence the equation to the line through A perpendicular I
BC is (Art. 44)
The equation to AC is
V
u x '
to the lie
hence the equati
AC is
■■(*)•
licular
=>(«—>
through B perpendicular
(«).
The line through C perpendicular to AB will be parallel
to the axis of y, and its equation will be (Art. 15)
■-«' (6)-
Now at the point of intersection of (5) and (6) we have
x = x\ y = ~^,{x'-a);
and as these values satisfy (i), the line represented by (*
passes through the intersection of the lines represented by
(5) and (G).
The lines drawn through the middle points of the sides of
a triangle respectively perpendicular to those aides meet in
a point.
X
PEOPERTIES OF A TRIANGLE. 51
The equation to the line through D perpendicular to PC is
y' x' — af a + x'\ , x
The equation to the line through E perpendicular to GA is
Hf"?(-l) (8) -
The equation to the line through F perpendicular to AB is
*=2 (»).
Now at the point of intersection of (8) and (9) we have
— a _ 2.' — ?L ( a _ ?L\
~2' y ""2 y' \2 2J ;
hese values satisfjr (7) ; hence the lines represented by (7),
8), and (9), meet in a point.
Let us denote by P the point of intersection of the three
ines in the first proposition, by Q the point of intersection of
he three lines in the second proposition, and by R the point
>f intersection of the three lines in the third proposition ; we
vill now prove that P, Q, and 22 lie in one straight line.
The co-ordinates
v'
of Pare a> = J(a>' + a), y = ^;
of Q are x = x, y=^n(p — #') I
e -n & y' rf ( a — x ')
oi M are x = - , v = — — — - — r— ' •
2 ^ 2 2y
Hence the equation to the line passing through P and Q is
y-8- y-i^a) ( g — H (10) -
4—2
-?L=t
52 EXAMPLES ON THE STRAIGHT LINE.
In this equation put x = - , then
—(---I
«■(«
3 /o a:"
Hence the point B is on the line represented by (10);
the co-ordinates of It satisfy (10).
*
the
1. Find the equations to the lines which pa3s thrf
the following pairs of points:
/(l) (0, lS, and'(l,-a).
I (2) (2, 3), and (2, 4).
(3) (1, 1), and (- 2, - 2).
^(4) (0,-a), and (0, -b).
2. Find the equations to the lines which pass througl
point (4, 4) and are inclined at an angle of 45° to the
y = 2x.
3. Find the equations to the lines which pass th.ro
the point (0, J), and are inclined at an angle of 30 u to
line y + x=2.
4. Find the equations to the lines which pass thro
the origin and are inclined at an angle of 45 to the
EXAMPLES ON THE STRAIGHT LIKE. 53-
5. Find the equations to the lines which pass through
the origin and are inclined at an angle of 60 to the line
6. Find the angle between the lines $e+y = l, y = «+2;
also find the co-ordinates of the point of intersection.
7. Find the angle between the lines a? + yV3 = and
x-y \/3 = 2.
8. What is the angle between se + 3y = l and a; — 2y=l?
9. Find the equations to the lines passing through a
given point in the axis of x> and making an angle of 45°
with the axis of a?.
10. Find the equation to the line which passes through
1 the origin and is perpendicular to the line x+y = 2.
11. Find the perpendicular distance of the paint (1,-2)
from the line x + y — 3 = 0.
12. Find the length of the perpendicular from the point
x 1/
la, b) on the line - + t = !•
v ' a o
13. Find the co-ordinates of the point of intersection of
the lines - + f = 1 and % + ^ = 1.
a b b a
14. Find the equation to the line which passes through
the point (a, 6), and through the intersection of the lines
abba
15. Shew what loci are represented by the equations
(1) rf + *■-<>, (2) rf-y' + O,
(3)^+0^=0, (4) xy = 0,
{5)\?+y t +a , = 0, (6) a;(y-a)=0.
7,
EXAMPLES ON THE STRAIGHT LINE.
I.
Interpret
(l) (*-«)(y-JH0,
{2) (a-oj'+ty-aj'-o,
(3) ( x -y + a)'+(x + y-a)' = 0.
I 17. What straiglit lines are represented by the equntiot
f - ixy + 3^ = ?
18. Shew that %tf - Sxy - 3x* + 30x — 27 = represent!
two straight lines at right angles to one another,
19, Find the equations to the diagonals of the four-Bide!
figure, the sides of which are represented by the equations
t!n
20. ABCDEF is a regular hexagon ; take A for
origin, AB a3 axis of x, and a line through A perpendiculir
to AB as axis of y; find the equations to all the lines joining
the angular points of the hexagon.
21. Given the co-ordinates of the angular points ofi
triangle, find the equation to the line which joins the middi
points of two sides.
22. Find the tangent of the angle between the lines
y — mx = and my + x = 0,
when referred to oblique axes.
23. Shew that whether the axes be rectangular or oblique
the lines y + x = and y — x — are at right angles.
24. Given the lengths of two sides of a parallelogram
and the angle between them, write down the equations to the
two diagonals and find the angle between them ; taking
of the corners as origin, and the two sides which meet in that
corner as axes.
25. In the figure to Art. 76, take BA and BC as the
of x and y; suppose BA = a t BC = c; and let h, k be the
co-ordinates of D; then form the equations to AG, BD,
AD, CD.
EXAMPLES ON THE 8TRAIQHT LINE.
26. With the notation of the preceding question, find
lie co-ordinates of the middle point of AG and those of the
middle point of BD, and form the equation to the line passing
through these two points.
27. With the same notation find the co-ordinates of the
middle point of EF, and thus shew that this point lies on
the line joining the middle points of A and BD.
28. If-+^ = 1 and -, + J-, = 1 be the equations to two
iea, which with the co-ordinate axes (rectangular or ob-
lique) contain equal areas, and x, y' be the eo-ordinates of
the point of their intersection ; shew that
V 6 - 6 '
29. What points on the axis of a: are at a perpendicular
distance a from the line - + \= 1 ?
a o
30. Form the equation for determining the abscissa of a
point, in the straight line of which the equation is - -f £ = 1 ,
whose distance from a given point (ct, j3) shall be equal to a
given line c. Shew that there are in general two such points,
and in the particular case in which those points coincide
31. Find, the tangent of the angle between the two lines
represented by the equation
Ay' + Bxy + Cx* = 0.
32. Find the points of intersection of the straight lines
x + 2y — 5 = 0, 2x + y — 1 = 0, and y — x — 1 = ; and shew
that the area of the triangle formed by them is - ,
33. The area of the triangle formed by the straight lines
y = x tan a, y = x tan £, # = a; tan 7 + c,
c* sin (a — /9) coh'y
2 sin (a — 7) sin (/3 — 7) '
56 EXAMINES ON THE STRAIGHT LINE.
34. Given the equations to two parallel straight
find the distance between them.
i 35. Determine the angle between the lines
- = 4 cos + 3 sin 0, - = 3 cos 0-4 sin 8.
36. Interpret F (6) = 0; for example, sin 30 =
37. If the axes be inclined at an angle w, the conditio!
that the lines
Ax + By+C = 0, A'x + B'y+C'^O,
may be equally inclined to the axis of x in opposite dine
Uons is
B B
A + A' =
•2 COS (,;
38. In the preceding question, if besides being equal!;
inclined to the axis of x the lines pass through the origii
and are perpendicular to one another, the equation to tb
straight lines i3
X s + 2zy cos oi + y' cos 2a> = 0.
39. Two parallel line3 are drawn at an inclination S\
the axis of x through the two points whose co-ordinates si
a, b, and a, b'\ shew that the distance between these lint
is (J' — J) cos 8 — (a' — a) sin 8. Hence determine the rec
angle whose sides pass through four given points, and. whoi
area is given.
40. A square is moved so as always to have the tn
extremities of one of its diagonals upon two fixed lines i
right angles to each other in the plane of the square ; she
that the extremities of the other diagonals will at the san
time move upon two other fixed straight lines at right angli
to each other,
41. AB and BG are two lines perpendicular to eac
other, A is a fixed point, B moves along a given, rigl
line, and AB to BG is a given ratio; determine the loci
of G.
EXAMPLES ON THE STRAIGHT LINE. 57
c 42. OX and OF are fixed lines meeting in any angle;
& line of given length slides along OX, and another line
of given length slides along OY. Find the locus of a point
^vhich is so taken that the sum of the areas formed by joining
it to the ends of the moving lines is constant.
43. Shew that the lines FG f KB, AL, in the figure to
Uuclid I. 47, meet in a point.
i 44. If upon the sides of a triangle as diagonals, paral-
lelograms be described, having their sides parallel to two
given lines, the other diagonals of the parallelograms will
meet in a point.
45. If from a fixed point a straight line be drawn
OA.BCD... meeting in-^, B, (7,2),... any given fixed straight
Jines in one plane, and if
JL_J_. 1 1
" OX" 0A + U2 + W +m "
i X "being a point in 0A f the locus of X is a straight line.
46. Shew that the area of the triangle contained by the
axis of y and the lines
ft
IS
fc ■
to ,
1
y = m x x + c l9 y = rnp + c %>
fa-*.)'
47. Determine the area of the triangle contained by the
lines
y = m x x + c x , y^mp + Ci, y = mjc + c % .
48. The area of the triangle formed by the three straight
lines
he r ac ab
y = ax — , # = &»-—, y=*cx-
1S
8
CHAPTEE IV.
STRAIGHT LINE CONTINUED.
65. We have seen that each of the equations
Ax + By + (7 = 0, A'x+By+C' = 0,
represents a straight line. We will now interpret the
tion
Ax + By+ C+\(A , x + By + C') = (1),
where X i3 some constant quantity.
I. Equation (1) must represent some straight line, became
it is of the first degree in the variables x, y. (Art. 16.)
II. The line represented by (1) passe3 through the inter-
section of the lines represented by
Ax +By + C=0 (2),
A'x + B'y+C'^O (3).
For the values of x and y which satisfy simultaneously (3)
and (3) will obviously satisfy (I 1 - *
(2) and (3) intersect lies on (1).
III. By giving a suitable value to the constant X the
equation (1) may be made to represent any straight line which
passes through the intersection of (2) and (3).
For let x lt y i denote the co-ordinates of the point of inter-
section of (2) and (3) ; suppose any line drawn through this
point, and let a:,, y 3 be the co-ordinates of another point in it
Now we have already shewn in II. that the line (l)
through (x t , y t ) ; we have therefore only to prove that by
giving a suitable value to X the line (1) can be made to pasi
I
INTERPRETATION OF AN EQUATION.
through (a;,, yj, 'because two straight lines which have two
common points must coincide. Substitute x x , y t for x and y
respectively in (1), and determine X ao as to satisfy the equa-
Thus
tion.
Ax, + By, +C
■
A'x^+B'y^+V'
Now use this value of A. in (1); then the equation
Ax + a, + 0-^±^±§.(4'x + B;+ C") = 0...(4)
represents a straight line passing through [x 1 ,y i ) and {a;,, y,).
"We have thus proved that by giving a suitable value to X,
the equation (1) will represent any straight line passing
through the intersection of (2) and (3).
t 66. The preceding article is very important, and com-
. monly presents difficulties to beginners. The student should
not leave it until he is thoroughly familiar with the three
s propositions which are contained in it. The first proposition
is obvious. To prove the second proposition the student may,
if he pleases, actually find the values of x and y which satisfy
simultaneously Ax + By+C=0, and Ax + B'y + 0' = 0,
and convince himself, by substituting these values, that they
| do satisfy Ax+By + C + X (A'x + B'y + C) = 0. There is,
I however, no necessity for solving the first equations, because
it is evident that values of x and y which make Ax + By+ C
and A'x + B'y + C vanish simultaneously must make
Ax + By + 0+X (A'x + B'y + G') vanish, because they make
each of the two members of the expression vanish. The third
proposition of the preceding articfe is usually the most dif-
ficult — the student is apt to think it needs no demonstration.
It may be obvious, however, that by giving different values
to X, different lines are represented, and that we can thus
obtain as many lines as we please, but this does not shew
that we can by a suitable value of X in (1) represent any
line passing through the intersection of (2) and (3).
ABRIDGED NOTATION FOB STRAIGHT LINES.
For example, if the straight linea (2) and (3) be DSEiaA
FSO respectively, it might have happened that all the linn
represented by (1) fell within the angle FSD and
within FSE. It requires to be proved then that by giving to
X a suitable value in (1) we can obtain the equation to an;
line through S.
67. It is often convenient to denote by & single symbol
the expression which we equate to zero in our investigations
in this subject; for example, in Art. 51 we have used the
symbol a as an abbreviation for a; cos a + y Bin a— p. In
like manner we may denote such expressions as Ax + By+C,
. by single symbols, as u, v,..
Now it will be seen that the demonstration in Art, 65 applies
to any form of the equation to a straight line as well as to
the form Ax + By + 0=0 which we have used. Hence the
result may be enunciated thus : — if u — and v — be the
equations to two straight lines, and X. a constant quantity,
the equation w-f-Xv = will represent a straight line paaaillg
through the intersection of the two lines; and by giving a
suitable value to X, the equation will represent any straight
line passing through the intersection of the two liues.
68. If u = and v = be the equations to two straight
lineB, then as we have shewn, u + Xv = will represent a
straight line passing through their intersection; it is sometimes
convenient to use the more symmetrical form lu+ mv = 0,
where I and m are both constants. It is obvious that what has
been said respecting the hrst form applies to the second ; in
.
ABRIDGED NOTATION FOR STRAIGHT LINES.
t must be remembered throughout this chapter that I, m,
,... X,... are constants, though for shortness we may omit to
Eate it specially in every article.
59. Similarly if w = 0, v = 0, w = 0, he the equations to
3 straight lines, and I, m, n be constants, the equation
lu + mv + nw = (1)
will represent a straight line. Moreover, by giving suitable
values to I, m, n we may in general make this equation re-
present any straight line whatsoever. For suppose we wish
this equation to represent the straight line passing t hrough
(a;,, y,) and (x„, y j. E et u t , i \, v.\ di-notc the valiu-s of it, r,
~ irres pectively when we put .i\ for x and ^for^; and let w,,
■ v„ w, be the respective values when x % and y, are put for x
and y respectively. Then determine the values of -r and -r
j from the equations
lu l +mv, + niB t = 0,
suppose we thus find
m _n n _v
substitute these values in the equation
w+-fV+-fW = 0,
which represents the bine passing through the points (x t , y,)
and (a;,, yj.
We have said above that the equation (1) can in general
be made to represent any straight line, because there arc
exceptions which we now proceed to notice.
- (If
62 ABRIDGED NOTATION FOE STRAIGHT LINES.
"When the lines represented by w = 0, »=0, and to
me et in a point , the equation (1) represents a line wi
necessarily passes through that point. For since the thnt
given lines meet in a point, w, «, and w vanish simultaneous';
at that point; therefore lu + mv + nw also vanishes at thi
point, so that the line represented by equation. (1) paaset
through that point.
When the three_ give n_ _l_ines_ are _pj,ra]iej the equation
u = 0, v = 0, w = (Twill be of the form
Ax + By+G l = 0, (<■) L * — I
Ax + By+ C t = 0, |»)
Jm + Qf+d.-QtQi ,
and thus equation (1) may be reduced to
. J _ , C,+ C,+ <7, „
A * + B a + ] + m \ n • -»■
and this equation represents a line parallel to the given lines.
Thus if the three given lines meet in a point or in
parallel, equation (1) will not represent any straight line ; f«
the line represented by equation (1), in the former case passes
through the point in which the given lines meet, and in the
latter case is parallel to the given lines.
We may shew that there is no other exception. For the
only case in which the general investigation can fail is when
\, fi. and v all vanish, that is, when
£,«,— H),M, =0 ]■ (2).
u,t>,=0j
We shall now prove that when equations (2) are satisfied,
the three given line3 either all meet in a point or are parallel.
■ ABHIDGED NOTATION FOE STRAIGHT LINES. 63
i First suppose that the points (x„ y,) and (a;,, y,) are not
:on any of the three given lines ; so that none of the quan-
f? titles u,, v.,
vanish.
From the first of equations (2) we have
hence by Art. 47 it follows that the ratio of the perpendicu-
lars from (x , y,) and (a;., y,) on the line v = 0, is the same as
the ratio of the perpendiculars from the same points on the
line w = 0. Hence it will follow geometrically either that
— the lines v = and w = are both parallel to the line joining
(as,, y,) and (as,,y,), or else that these three lines meet in
~ a point. Similar results follow from the second of equations
(2), and from the third of equations (2). Hence in this case
if equations (2) are satisfied, the three given lines either meet
in a point or are parallel.
Next suppose that one of the two given points is situated
on one of the three given lines ; suppose for example that
w, = 0. Then from the first of equations (2) it follows that
*- either w,=0 or to, = 0. Suppose we take v =0. Then
from the second and third of equations (2) we deduce
* either that u = or else that w a = and «, = ; ia the
* former case the three given lines all pass through the point
- {a:,, y,} ; in the latter case the lines v = and w> = both
~ pass through the two points (a:,, y t ) and (as,, y^ t that is,
two of the given lines coincide so that all three will reduce
either to two intersecting lines or to two parallel lines. Sup-
pose we take wj„ = in conjunction with w l = 0. Then the
line to = passes through the given points (a;,, yj and (ar g , yj.
From the third of equations (2) we nave
and thus the lines « = and v = either meet on the line
joining the points {a;,, yj and (a;,, yj, or are parallel to this
line ; that is, the lines w = 0, v = 0, and w =0 either meet in a
point or are parallel.
CI
ABRIDGED NOTATION FOR STRAIGHT L1NEJ
70. Let a = 0, 8 = be the equations to two lines
pressed in terms of the perpendiculars from the origin
their inclinations to the axis of x (see Art. 50), so that a i
abbreviation for a; cos a + y sin a — p lt and ,9 is an at
viation for x cos ,9 + y sin fi — p, ; we proceed to shew
meaning of the equations a — 8 = and a+/9 = 0.
Let
SA be the line a = 0,
SB 9 = 0;
let SC bisect the angle A 8B, and SB bisect the supplen
of A8B; the angle BSG is therefore a right angle. 1
any point P in SC and draw the perpendiculars PM, Ph
SA, SB respectively. If x, y be the co-ordinates of P,
length of PM is a by Art. 54, and the length of PN i
Since SC bisects the angle A SB, PH=PN; therefor*
any point in SO we have 8 = a ; that is, the equatit
80 £
a-B.
Similarly, the equation to SB is
EXAMPLES OF ABRIDGED NOTATION. 65
Thus a~/9=0 and a+£=0 represent the two lines which
188 through the intersection of a = and y9 = and bisect
t6 angles formed by these lines.
71. The student must distinguish between the lines
— )8 = and a + £ = 0; the following rule may be used:
<e two lines a=0, £=0, will divide the plane in which they
* into four compartments ; ascertain in which of these com-
jrtments the origin of co-ordinates is situated; a — #==0
sects that angle between a=0 and £ = in which the
Lgin of co-ordinates lies. This is obvious from the investi-
"taon in the preceding article and the remarks in Arts.
, 54.
The equation a + X/8 = represents a line such that X is
unerically equal to the ratio of the perpendicular from any
•int of it on a = to the perpendicular from the same point
. £ = 0. If X is positive the line a + X# = lies in the
cne two of the four compartments just alluded to as the
le a + ft = ; if X be negative the line a + \/3 = lies in
^ same two compartments as the line a — £ = 0. From
e figure to Art. 70 we see that PM— PS sin PSM and
tf^PSsinPSN: hence X or -~vr= . — ^ovf; that is, X ex-
7 PN amPSN '
resses the ratio of the sine of the angle between a = and
■f \£ =0 to the sine of the angle between ft = and a + A$=0.
72. We shall continue to express the equation to a straight
ine by the abbreviation a = when the equation is of the
)nn a?cosa + y sin a— p=0; when we do not wish to re-
trict ourselves to this form, we shall use such notation as
= 0, v = 0, w' = 0,
Let u = 0, v = be the equations to two lines, the axes
eing rectangular or oblique; then u — Xv = and u + \v =
^present two lines passing through the intersection of the
rst two. Suppose, as in Art. 70, that SA, SB are the first
wo lines and 8 C, SD the second two ; then will
sin GS A _ sin PSA •
sinC£5~sinZ>£S'
T. C. S. * ^>
66 EXAMPLES OF ABRIDGED NOTATION.
For by* Art, 57 it appears that if p be the perpendicular
from a point (x, y) on the line u = 0, tnen p = /*v, where p i
a constant quantity; similarly if p denote the perpendicular
from the same point on v = 0, then p=fi'v, where y£ i& a con-
stant quantity. Hence the equation u — Xt; = 0, or 2.— -£L =
/* /*
shews that — , = -%-; thus we see that numerically withorf
regard to algebraical sign
sin C8A _ \fi
sin C'&# ~ y *
sin 2)&4 X/a
Similarly,
ainl> SB ~ p"
mGBA^^DBA
sin C'tfjB sin D&B '
sJzzX" 73. We will apply the principles of the preceding article!
to some examples.
Let a=0, £ = 0, 7 = be the equations to three lines
which meet and form a triangle, and suppose the origin i
co-ordinates within the triangle ; then the equations to tfc
three lines bisecting the interior angles of the triangle a$
by Art. 70,
£-7 = 0.. .(1); 7-a = 0...(2); a-£ = 0...(3).
These three lines meet in a point ; for it is obvious iirt
the values of x and y which simultaneously satisfy (1) anl
(2) will also satisfy (3).
Again the equations to the three lines which pass through
the angles of the triangle and bisect the angles supplemental
to those of the triangle are
£ + 7 = 0.. .(4); 7+a = 0...(5); a + £ = 0...(6).
It is obvious that (3), (4), and (5) meet in a point; simi-
larly (5), (6), and (1) meet in a point; so likewise (4), (6),
and (2) meet in a point.
In all our propositions and examples of this kind, we shall
always suppose the origin of co-ordinates within the triangle,
unless the contrary be stated. *%
{
EXAMPLES OP ABRIDGED NOTATION. 67
74. If a = 0, fi = 0, 7 = be the equations to three lines
hich form a triangle, then any line may be represented by
a equation of the form la -f mfi + ny = ; for the exceptional
uses noticed in Art. 69 cannot occur here.
*
Let a, b, c denote the lengths of the sides of the triangle
hich form parts of the lines a = 0, j8 = 0, 7 = respectively.
ake any point within the triangle and join it with the
tree angular points ; thus we obtain three triangles the areas
' which are respectively — — , — — , and — -£. Hence
A A A
aa+b/3+cy = & constant;
ie constant being in fact twice the area of the triangle taken
egatively.
This result holds obviously for any point within the triangle
stermined by a = 0, £ = 0, 7 = 0. It will be found on ex-
nining the different cases which arise that it is also true for
iy point without the triangle. Hence it is universally true.
Suppose we require the equation to a line parallel to the line
la + m/3 + ny = 0.
Ids required equation may be written
la + mfi + ny + k = 0,
rhere h is a constant. (Art. 38.)
Or, since aa + b/3 + erf is a constant, the required equation
lay be written thus,
la + m0 + ny + k' (aa + bfi + cy) = 0,
here 1c is a constant. n
75. The lines represented by the equations u =• 0, v = 0,
> = 0, will meet in a point, provided lu+mv + nw is iden-
cally = ; 7, m 9 n being constants. For if lu + mv + nw =
lenticaMy, we have
lu + mv ,
w — always.
* % n J
68 EXAMPLES OF ABRIDGED NOTATION.
«
Hence the equation w = may be written
lu + mv
n
= 0,
that is, the line w = is a line passing through the inter-
section of u = and v = ()•
76. The following example will furnish a good exercb
in the subject.
MM
B A E
Let ABCD be a quadrilateral ; draw the diagonals AC,
BD ; produce BA and CD to meet in E, and AD and BG to
meet in F; join EF 9 forming what is called the third diagmd
of the quadrilateral. Suppose
m = 0, the equation to AB, (1),
v = 9 BC, (2),
w = 0, CD, (3).
We propose to express the equations to the other lines d
the figure in terms of u, v, w, and constant quantities. As-
sume for the equation to BD
lu — mv = (4),
and for the equation to CA
mv — nw = „ (5).
These assumptions are legitimate, because (4) represents
some line passing through B, whatever be the values of the
EXAMPLES OP ABRIDGED NOTATION. 69
constants I and m ; by properly assuming these constants, we
-may therefore make (4) represent BD. Also (5) represents
same line through C> and by giving a suitable value to n, we
may make it represent CA. We may if we please suppose
one of the three constants l> m, n, equal to unity, but for the
sake of symmetry we will not make this supposition. The
equation to AD is
lu— mv + nw = (6);
ibr (6) represents a line passing through the intersection of
lu — mv = and w = 0, tnat is, a line through D ; also (6)
represents a line passing through the intersection of w =
and mv — nw = 0, that is, a line through A. Hence (6) re-
presents AD. The equation to EF is
lu+nw = (7);
for (7) obviously represents some line through E, and since
lu + nw = lu — mv -f nw + mv, (7) represents some line through
JB\ Hence (7) represents EF.
Let Q be the intersection of AG and BD. Thfc equatiQn
to EG\&
lu—nw=0 (8);
for (8) represents a line passing through the intersection of
(1) and (3), and also through the intersection of (4) and (5).
The equation to FO is
lu — 2mv + nw=*Q„ , ,. (9) ;
for (9) represents a line passing through the intersection of
(4) and (5), and also through the intersection of (9) and (6).
Suppose BD produced to meet EF in H, and A G and
EF produced to meet in K\ then it may be shewn that the
equation to
A H is 2lu — mv + nw « 0,
that to CZF is tnv + nw — Q f
KB is lu + mv = 0,
KD is lu — mv -f %nw ** 0.
We have introduced this example, not on account of any
importance in the results, but as an exercise in forming the
equations to lines. We proceed to another example.
70 EXAMPLES OP ABRIDGED NOTATION.
77. If there be two triangles such that the lines joininf;
the corresponding angles meet in a point, then the intersec-
tions of the corresponding aide3 lie in a straight line.
Let ABC be one triangle, A'BC the other triangle ; let S
be the point in which the lines AA', BB 1 , CG' meet. Let
the equation to BChsu = 0, to CA w = 0, and to AB w?=0.
3 for the equation to
SG' Tu + mv + nw-
and to C'A' lu +m'v + nw =
•w,
-M-
It 13 shewn in Art. 69 that the equation to B'C may be
written in the above form, and by the method of that article
it may be shewn that by giving suitable values to the con-
stants I, m, we may make (2) represent C'A'. We will non
prove that the equation to AB' may be written in the form
lu + mv + n'w = Q (3).
The constant ri may be obviously determined, ao as
make the line represented by (3) pass through A' ; let ri be
so determined ; it remains to shew that the line (3) will p
through B\ From (1) and (2) it follows that the equation
(l'-l)u + (m-m')v=0 (4)
represents some line through G'\ but (4) obviously represents
a line passing through the intersection of BC and CA. Hence
(■I) is the equation to CG'.
PROPERTY OP TRIANGLES. 71
Again, the line represented by (3) by supposition passes
through A'; hence from (2) and (3) we see that
(m'~ m)v + (n — ri) w = (5)
is the equation to AA\
The equation
(Z'-Z)w-f (tt-n')w>=0 (6)
represents a line passing through the intersection of BO and
^AH, that is, through B; and from (4) and (5) it follows that
•this line passes through the intersection of CC and AA\-
that is, through S. Hence (6) is the equation to SB.
Now from (1) and (3) it follows that the lines represented
T>y these equations meet on the line (6). Hence (3) is the
equation to A'B.
The required proposition now easily follows: for the line
represented by
lu + mv + nw=zO (7)
passes through the intersection of BO and B'C\ of CA and
CA', and ot AB and A'B'; that is, these three intersections
are in the same straight line.
Conversely, if there be two triangles such that the inter-
sections of the corresponding sides lie in a straight line, then
the lines joining the corresponding angles meet in anoint.
To prove this we may begin with the equations to Bu, CA,
AB, B'C, CA' as before, and assume (3) as the equation to
some line through A'. Then (7) will represent the line pass-
ing through the intersection of BO and BC, and of CA and
C'A'; now (3) is the equation to a line passing through the
intersection of AB and (7) ; hence (3) must be the equation to
A'B. Then from the form of (1), (2), and (3), it follows im-
mediately that CC 1 passes through the intersection of AA
and BE.
It may be shewn also that the equation to the line which
passes through the intersection of AB and A' C, and of AC
and A'B is
lu + m'v + nw-0 (8).
And the intersection of (8) with BO will lie on the line
l'u + m'v + riw = (9).
72 EXAMPLES ON THE STRAIGHT LINE.
Similarly the line joining the intersection of BA and B<
with the intersection of BC and BA' will meet CA on
And also the line joining the intersection of CA and CI
with the intersection of CB and CA' will meet AS on (9).
78. The equation u + Xv = represents a straight lira
passing through the intersection of the lines w=0.
Hence if there be a series of straight lines the equations a
which are all of the form u + \v = Q, and differ merely !.
having different values of the constant X, all these lines pas
through a point, namely, the intersection of u — Q and u = Q.
1. Find the equation to the straight line passing throng
F the origin and the point of intersection of the lines
?+|-l, 3 + jUli
a o a b
2. A, A are two points on the axis of x, and B, B m
■ that of y, at given distances from the origin ; AB and A3
intersect in P, and AB' and A'B in Q ; find the equation to
the straight line PQ, and shew that the axes are divided
harmonically by it.
3. If a. = 0, £ = 0, 7 = be the equations to the sidi*
of a triangle ABC opposite the angles A, B, C, prove thai
a sin A — sin B=Q is the equation to the straight Use
bisecting AB from C.
4. Prove by means of such equations as that given in the
preceding question the first proposition in Art. 64.
5. Shew that a. cos A—0 cos B= is the equation to the
perpendicular from C on AB.
6. Hence prove the second proposition in Art. 64.
EXAMPLES ON THE STRAIGHT LINE* 73
7. If a, b, c be the lengths of the sides of a triangle
pposite the angles A, B, C, respectively, prove that
a cos .4 — £cos-B+- (sin B cos A — sin A cos -B) =
the equation to the line which bisects AB and is perpen-
jcalar to it. The equation may also be written
(. asinl?sin(7\ A f Qt JsinCsin^N ^ _
a+ —^-s — - A — }cQaA-(p + — — -. — =,— cos -5=0.
2 sm A J \ 2 sin B J
8. Hence prove the third proposition in Art. 64.
9. Interpret the equation aa + fy3 = 0.
10* Shew that aa + bfi — 07 = is the equation to the line
rbich joins the middle points of AC and BC.
11. Shew that
aco8-4 + £cos5— ycos (7=0
9 the equation to the line which joins the feet of the perpen-
liculars from A on BC, and from J? on AC.
12. If lines, be drawn bisecting the angles of a triangle
Aid the exterior angles formed by producing the sides, these
ines will intersect in only four points besides the angles of
he triangle.
13. If t* = 0, v = 0, w = be the equations to three
traight lines, find the equation to the line passing through
lie two points
"t — — ~ " « and 7? -*• ~ ~i — —7 .
£ m n b m n
14. Find the equation to the straight line passing through
tie intersections of the pairs of lines
2au + bv + cw — 0, bv — cw = 0;
nd 2bu + av + cw = 0, av — cw = 0.
>\
( 76 )
NATES.
g articles thai i
THANSFuJl.MATIuN OF C'O-OKDINATES,
79. We have seen in the preceding i
general equation to a straight line is of the form y=mns
but that the equation takes more simple forms in partial
cases. If the origin is on the line the equation becue
g = mx; if the axis of x coincides with the line, the eqwW
becomes y = 0. In a similar manner we shall 8ee aa '
proceed that the equation to a curve often assumes a mt
or less simple form, according to the position of the oiij
and of the axes. It 13 consequently found convenient
introduce the propositions of the present chapter, which enil
us when we know the co-ordinates of a point with respect
any origin and axes, to express the co-ordinates of the si
point with respect to any other given origin and axes.
will be seen that these propositions might have been pli
at the end of the first chapter, as they involve none of
results of the succeeding chapters.
80. To rlmiiijr the origin of co-ordinates without cAait
the direction of the axes, the axes being oblique or rcctangi
CHANGE IN DIRECTION OF RECTANGULAR AXES.
77
Let OX, OF be the original axes; O'X', OY the new
so that OX' is parallel to OX, and OY' to OY.
rt h, k be the co-ordinates of 0' with respect to 0. Let P
«ny point ; x, y its co-ordinates referred to the old axes ;
y' its co-ordinates referred to the new axes.
Let T O produced cut OX in A ; draw PM parallel to Y
Stfeting OX f in N; then
0^ = 4, -40'=£;
x~ 0if=^Of+ 0A=0'N+ 0A=x' + h,
y = PM=PN+NM=PN+AO'=y' + Jc.
Hence the old co-ordinates of P are expressed in terms of
1 new co-ordinates.
81. To change the direction of the axes without changing
origin, both systems heing rectangular.
M n
Let OX, OY be the old axes; OX', OY' the new axes,
joth systems being rectangular ; let the angle XOX' = 0.
[jet P be any point; x, y its co-ordinates referred to the old
ixes ; x', y' its co-ordinates referred to the new axes. Draw
PM parallel to OY, PM' parallel to OY', M'N parallel to
OY, and M 'R parallel to OX.
Then x = OM = 0J\T- JQT=s OZ\T- jlf 'J2
= 0Jf ' cos XOX' - PJf' sin M 'PR
**x' co&0—y' sin0;
78 TO CHANGE THE DIRECTION
y = PM = RM + PS =- M 'N+ PR
= x' sin0 +y' cos0.
Hence the old co-ordinates of P are expressed in tern
its new co-ordinates.
82. In the preceding article 6 is measured from the m
tive part of the axis of x towards the positive part of the Hi
of y; therefore if in any example to which the formula i
applied, OX' fall on the other side of OX, must be ci
sidered negative.
From the formulae of the preceding article, we see that
a* + y* = x'* + y";
this of course should be the case, since the distance OPist
same whichever system of axes we use. 7y
83. To change the direction of the axes without chang\
the origin, both systems being oblique.
Let OX, OY be the old axes ; OX', OY' the new ax
Let (XY) denote the angle between OX, OY; and lei
similar notation be used to express the other angles wh:
OF OBLIQUE AXES. 79
m formed by the lines meeting at 0. Let P be any point ;
y its co-ordinates referred to the old axes ; x\ y' its co-
xlmates referred to the new axes. Draw PM parallel to
> F, and PM' parallel to OY'; from. P and M' draw PL,
*'N perpendicular to OY; from M ' draw M'R perpendicular
> PL. Then
x=OM, y = PM;
r V=oa/', y'=PM:
Now PL = perpendicular from if on 0F=ajsin (XY),
«o PL = BL + PB = M'N+PB
= OM' ainX' OY+PM' sin F'OF
=*afmn{X'Y)+tfmn(Y'Y);
.-. a?sin(Xr) = a? , sin(X l r)+y , sin(r'r).. (1).
Similarly by drawing from P and M ' perpendiculars on
1ST we may prove that
ysin(rX)=a/sin(X'X)+y'sin(r'X) (2).
Equations (1) and (2) express the old co-ordinates of P in
arms of its new co-ordinates; (YX) and {XY) denote the
%me angle, but we use both expressions for greater sym-
letiy.
Let X0X' = a, X0F' = /3, XOY=a>; then (1) and (2)
ecome
x sin co = x am (© — a) + y' sin (© — £) (3),
y sinG>=a/sina -f y'sin# (4).
84. Two particular cases of the general proposition in the
receding article may be noticed.
If the original axes are rectangular © = — , and the equa-
ions (3) and (4) become
x = x' cos a + y' cos £,
y = x' sin a + y ' sin #.
80
CHANGE OF ORIGIN AND DIRECTION.
If the new axes be rectangular /3 = - + a, and the eqiifr|
tions (3) and (4) become
x sin a> = x' sin (a> — a) — y cos (a> — a),
y sin co = a' sin a + y' cos a.
85. Suppose we require to change both the origin and
the direction of the axes; let x, y be the co-ordinates of a
point referred to the old axes; x', y' the co-ordinates of the
same point referred to the new axes; By Arts. 80 and IB |
we have
x = x x + A,
where h and h are the co-ordinates of the new origin referral
to the old axes, and
x t =
_ x' sin (co — a) + y' sin (to — ft)
sina>
__ x f sin a + y ' sin /8
^ "" sin ©
The expressions for x x and y t will simplify when one or
each of the systems is rectangular. (See Art. 84.) ^ /
86. The formulae which connect the rectangular and polar
co-ordinates of a point in the particular case in which the
origin is the same in both systems, and the axis of x coincide
with the initial line, have already been given. (S^e Art 8.)
The following is the general proposition.
To connect the polar and rectangular co-ordinates of a
point.
Let OX, Y be the rectangular axes ; let S be the p&
and 8A the initial line. Let A, k be the co-ordinates of &
referred to ; draw 8X' parallel to OX, and let the angle
ASX' = a.
POLAB ANP BECTAMOULAK C0-OBDINATES.
81
1 .•••.V ;,\ -/< „v ? ,.' .
V.--
.'1
•-•>
Tjet P be any point ; x, y its co-ordinates referred to the
«tangular axes; r, its polar co-ordinates. Draw PM,
<7 parallel to OY f the former cutting SX' in N 7 and join
JP; then
r = SP, = the angle PSA.
*d a?= 0(7+ CJf= 0(7+ BN
= A + rcos (0 + a) (1),
y = MN+PN=SC + PN
= & + rsin(0 + a) (2).
If a = we have
a? = A + rcos0 , (3),
y = & + rsin0 (4).
87. By means of the formulae of the present chapter we
hall sometimes be able to simplify the form of an equation ;
or example, the axes being rectangular, suppose we have
y 4 + * 4 + 6ay = 2 (1).
This equation represents some locus, and by ascribing
liferent values to x and determining the corresponding
'alues of y from the equation, we can find as many points
«f the locus as we please. The equation however will be
implified by turning the axes through an angle of 45°. In
he formulas of Art. 81 put — for 6 ; thus
T. C. S. ^
82 ■ EXAMPLES OP TRANSFORMATION
*' — y' x ' + y' m
Substitute these values in (1) ; thu3
(«/+y>' + (rf-y)* + 6(.tf»-jrT-8;
.-. 2 (as" + 6*V + j") + G (a* - y") 1 = 8,
or x' 1 + y" = 1 (J
Since (3) is a simpler form than (1), we shall find it ea
to trace the locus by using (3) and the new axes, than
Using (1) and the old axes. The student must observe
we make no change in the locus by thus changing the!
or the origin to which we refer it ; that is, equation
represents precisely the same assemblage of points as
for instance, the point for which x'=l and y'=0 is obvio
situated on the locus (3) ; now this point will by (2) haw
its co-ordinates referred to the old system
and these values satisfy (l), that is, this point is on
locus (1).
We may remark that we cannot alter the degree o
equation by transforming the co-ordinates. For if in
expression Ax*y? we substitute the values of x and y in I
of x' and y given in Arts. 80 — 84, we obtain
A (ax' + by + k)* (ex + ey' + h)*,
where a, b, c, e, k, k are all constant quantities; by expai
this expression we shall obtain a series of terms of the
A'x'iy' 1 , where y + S cannot be greater than a + (3. I
the degree of an equation cannot be raisedhj transforn
of co-ordinates. Neither can it be depressed; for if fi
given equation we could by transformation obtain one
lower degree, then by retracing our steps we should he
from the second equation to obtain one of a higher di
which has been proved to be impossible.
• OP THE CO-ORDINATES. 83
EXAMPLES.
.. Change the equation r* = a* cos 20 into one between
dy.
f. Shew that the equation 4xy — So? = a 8 is changed into
4y* =5 a*, if the axes be turned through an angle whose
ant is 2.
. Transform \Jx + *Jy = *Jc so, that the new axis of x
be inclined at 45° to the original axis.
;• The equation to a curve referred to rectangular axes
1 +Aay cot a — 4oa? = ; find its equation referred to
[ue axes inclined at an angle a retaining the same axis
*
. Shew that the equation a?y* = a (a? + y 8 ) will admit of
ion witE~respect to y if the axes be moved through an
> of 45°.
. If a?, y be co-ordinates of a point referred to one system
►lique axes, and x\ y' the co-ordinates. of the same point
•ea to another system of oblique axes, and
x = mx + ny\ y = mix 1 + n 'y 9
that
m 2 + m!* — 1 mm'
n* + ri* — 1 nra
/ •
( 84 )
CHAPTER VI.
THE CIRCLE.
88. We now proceed to the consideration of t
represented by equations of the second degree ; the s
of these is the circle, with which we shall commence,
Tojind the equation to the circle referred to any reck
axes.
N
M
X
Let G be the centre of the circle ; P any point oi
cumference. Let c be the radius of the circle ; a, b
ordinates of C; x, y the co-ordinates of P. Draw
parallel to OY, and CQ parallel to OX. Then
CQ* + PQ 2 =CP 2 ;
that is, (x-a)*+(y-b) 2 = c 2
or a? + y 2 - 2 ax - 2 b if + a 2 + b 2 - c 2 =
This is the equation required.
EQUATION TO THB QIRCLE. 85
ie following varieties occur in the equation.
Suppose the origin of co-ordinates at the centre of the
; then a =* 0, and b = ; thus (1) and (2) become
a?+y*~c* = Q (3).
• Suppose the origin on the circumference of the circle ;
the values x « 0, y «= 0, must satisfy (1) and (2) ;
ore
l relation is also obvious from the figure, when is on
rcumference; hence (2) becomes
a? + y*-2ax-2by**0 (4).
I. Suppose the origin is on the circumference, and that
ameter which passes through the origin is taken for the ^
►f x ; then b = 0, and a* = c 2i , hence (2) becomes
of + y*-2ax = (5).
milarly if the origin be on the circumference and the
>f y coincide with the diameter through the origin, we
% = 0, and b* = c* ; hence (2) becomes
x* + y'~2by=0 (6).
ence we conclude from (2) and the following equations,
the equation to a circle when the axes are rectangular
rays of the form
a? + y* + Ax + By+C=Q,
j A y B, G are constant quantities any one or more of
i in particular cases may be equal to zero.
>. We shall next examine, conversely, if the equation
a? + y* + Ax + By + (7=0 (1)
r s has a circle for its locus,
quation (1) may be written
(*+-)•+ (y +|).*-±±£ - C. (2).
86
TANGENT TO A CIRCLE.
I. If -4 2 + jB 8 — 4 C be negative, the locus is impossible* |
II. If ^l 2 +-B 2 -4<7 = 0, equation (2) represents a
A B
the co-ordinates of which are — — , — — . This point may I
considered as a circle which has an indefinitely small radios*
III. If A* + jB 8 — 4 C be positive, we see by coin
equation (2) with equation (1) of the preceding article thill
represents a circle, such that the co-ordinates of its centre
_^ -f, and its radius i(^ 2 + 5 8 -4C r .)*.
2 2
It will be a useful exercise to construct the circles ref
seated, by given equations of the form
a? + y* + Ax + By + (7=0.
For example, suppose
a? + y 9 + 4a?-8y-5 = 0,
or a + 2) 9 +(y-4) 2 =:5 + 4 + 16 = 25.
Here the co-ordinates of the centre are — 2, 4, and tk|
radius is 5.
Tangent and Normal to a Circle.
90. Def. Let two points be taken on a curve and*
secant drawn through them ; let the first point remain fixel
and the second point move on the curve up to the first; tkftl
secant in its limiting position is called the tangent to tk
curve at the first point.
91. To find the equation to the tangent at any point i
a circle.
Let the equation to the circle be
x* + y 2 = c
2
(!)•
Let x\ y be the co-ordinates of the point on the circle it
which the tangent is drawn ; and x", y the co-ordinates of 1
JANGENT TO A CIRCLE; 8?
a adjacent point on the circle. The equation to the secant
lrough (x, y') and (x'\ y") is
jt .J
• >
Now since (x' 7 y') and (x", y") are both on the circumfer-
nce of the circle,
: "by subtraction,
. x"*-x'* + y"*-y' 1 =0,
i (x" - x') (x" + x') + (y" - y') (y" + y') = ;
. y -y _ x + x * •
Hence (2) may be written
y-y'=-yr^>{x-x). (3).
Now in the limit when (x", y") coincides with (x } y'), we
lave x" = x\ and y" = y' ; hence (3) becomes
i *•& / i\ x / i\
y ~ y = "27^"" a? ) = "y (*-*)•
Thus the equation to the tangent at the point (x\ y') is
r
y-y'=~< (»-«0 (4).
This equation may be simplified ; by multiplying by y and
ransposing we have
xx' + yy' = x'* + y'*;
•'• xx ' ^* yy ^ ^ (**)•
92, The equation to the tangent can be conveniently ex-
pressed in terms of the tangent of the angle which the line
88
TANGENT TO A CIRCLE,
makes with tie axis of x. For the equation to the
at (a?', y') is
yy' + xx' = c",
or
x' , <?
a;
Let — r = w ; thus the equation becomes
* y
<? .
We have then to express -7 in terms of «i.
Now
and
a? =z — fny y
a^ + y'^c*;
.•. y" (I + m*) « c',
* V(i + »* )
Hence the equation to the tangent may "be written
y = mx + c V(l + w* 8 )-
Conversely every line whose equation is of this form is »
tangent to the circle.
93. The definition in Art. 90 may appear arbitrary to ike
student, and he may ask why we do not adopt that given ty
Euclid (Def. 2, Book in.). To this we reply that the defcfr
tion in Art. 90 will be convenient for every curve, which is not
the case with Euclid's definition. The student however cannot
at first be a judge of the necessity or propriety of any define
tion ; he must confine himself to examining the consequences
of the definition and the accuracy of the reasoning based
upon it.
We may easily shew however that the line represented ty
the equation
xx'+yy' = c* (1)
TANGENT TO A CIRCLE. 89
mwhez, according to Euclid's definition, the circle
^+y f =c t (2),
be point {x\ y') being supposed to lie on the circle. To find
™^6 point or points of intersection of the line and circle we
ibine the equations (1) and (2) ; substitute in (2) the value
p $r from (1), then
■e a?(aj , *+y ,2 )-2c 8 a?'aj + c 4 -cy f *0,
■r cV - 2c*x'x + cV*= ;
« . OS S2S Qp J
••• from (l),y = y'.
Hence (1) and (2) meet in only one point, the point (x' 9 y').
ttence (1) touches the circle according to Euclid's definition.
94. Also every line which meets the circle in one point
^olj is a tangent to the circle.
For suppose
*> he the equation 'to a circle and
y = wx + n
he equation io a straight line ; to find the points of inter-
ection of the line and circle we combine the equations ; thus
re obtain
(mto + nf + x***^
r (m 2 +l)a 2 +2»wa> + tt a -c 8 «*0
» determine the abscissae of the points. Now this quadratic
quation will have two roots except when
(«i f +l)(n f -cP)=mV,
hat is, when w f = c , (l+m a ).
90 TANGENT "TO A CIRCLE.
Hence if the straight line meets the circle it must roeel
in two points unless this condition holds, and then, by An',
the line is a tangent to the circle.
95. Instead of supposing one of the points on the cir
fixed and the other to move along the circle as in the d<
nition of Art. 90 we may suppose both to move along the cir
until they meet at some fixed point of the circle, and t
secant in its limiting position will be the tangent at ill
fixed point. For let (x 1 , y) and (x", y") denote then
moving points on the circle, and (x l , y t ) the fixed potfl
Then as in equation (3) of Art. 91, we shall have for !
equation to the secant
x' + x 1 . ,,.
y — y = - -, (x — X).
In the limit x' and x" each. = 3:,, and y and y" each =y l ,
we obtain for the equation to the tangent at (x 1 , y J
I which agrees with the former result.
96, If the equation to a circle be given in the form
(x - a)* + (y -b} 3 - C* = 0,
we may find the equation to the tangent at any point in tli
same manner as in Art. 91.
Let (x', y') be the point on the circle at which the taogfl
is drawn ; (a:", y") an adjacent point on the circle ; then
.
[x'-af+ti/'-bf-c'-O,
(»"-a)"+(y"-J)'-<P=.0i
.■.(*»-«)■- (*-«)■ + (y" - If - tf - })•_
or (a("-«0 (s" + «'-2«) + tf'-y')(2/"+y-2J)-0...(l]
Also tile equation to the secant through [x, y') and (x",y")
_ !->
,3i («-')•
NORMAL TO A CIRCLE. 91
By means of (1) this may be written
, x" + x'-2a, ,. /N
y^"" Y' + y-»» ( } ®'
' Now in the limit x" =*x' and y"=y' ; hence we have for
he equation to the tangent at (x\ y')
y~y'=-?!rzrE&- x ') (*)•
T * *»
This may be written
.: {x-*a)(x'-a) + (y-b)(y'-b)
= (aj'-a) 9 +(y-J) a = c 9 .. (5).
97. Dep. The normal at any point of a curve is a straight
ine drawn through that point perpendicular to the tangent to
he curve at that point.
98. To find the equation to the normal at any point of a rfk^^i
ircle. —
Let the equation to the circle be
*'+/ = c a (1),
,nd let x\ y' be the co-ordinates of a point on the circle, then
he equation to the tangent at that point is
xx'+yy' = <? f
x' c"
r y = ---,x + — .
y y
Hence the equation to a line through (x' 9 y') perpendicular
> the tangent at that point is
y'
92 TWO TANGENTS FROM AN EXTERNAL POINT.
Since this equation is satisfied by the valued a?=0, jr
the normal at any point passes through the origin of
nates, that is, through the centre of the circle* y
^eJts 99. From any external point two tangents can be dramb\
£Wk*a£ a circle.
r
Let the equation to a circle be
a'+y'-c' (i),
and let A, Jc be the co-ordinates of an external point. Sty-J
pose x, y' the co-ordinates of a point on the circle such twl
the tangent at this point passes through (A, k). The equatifltj
to the tangent at (a?', y') is
xx'+yy'=c* , ,. (2).
Since this tangent passes through (A, Jc)
hx+Jcy'=*c' (3).
Also since (x' t y*) is on the circle
*"+y" = c' (4).
Equations (3) and (4) determine the values of x' and}*
Substitute from (3) in (4), thus
, a [c* — hx'\* f
The roots of this quadratic will be found to be both possible
since (A, Jc) is an external point and therefore A* + Jc 9 create
than c a . To each value of x corresponds one value of y fcf
(3) ; hence two tangents can be drawn from any extend
point.
The line which passes through the points where these tan-
gents meet the circle is called the chord of contact.
100. Tangents are drawn to a circle from a given, extend
point ; to find the equation to the chord of contact .
Let A, Jc be the co-ordinates of the external point ; x v J {
the co-ordinates of the point where one of the tangents from
CHOBD OF CONTACT. 93
{h 9 Jc) meets the circle ; x r y t the co-ordinates of the point
"where the other tangent from (A, k) meets the circle.
The equation to the tangent at {x v yj is
asa? i+yyi ==c * • (*)•
Since this tangent passes through (h, k), we have
hx x +%«(?» (2).
Similarly, since the tangent at (a? 2 , yj passes through
1& t + 1cy % = <? (3).
Hence it follows that the equation to the chord of con-'
tact is
xh+yk=*<f. .. (4),
For (4) is obviously the equation to some straight line; also
-fliis line passes through (x x , y x ), for (4) is satisfied by the
values x = x v y=*y v as we see from ,(2); similarly from (3)
we conclude that this line passes through (x v yj. Hence
(4) is the required equation.
Thus we may proceed as follows in order to draw tangents
to a circle from a given external point — draw the line which
is represented by (4) ; join the points where it meets the circle
with the given external point and the lines thus obtained are
the required tangents.
101. Through any fioced point chords are drawn to a circle,
and tangents to the circle drawn at the extremities of each chord;
the locus of the intersection of the tangents is a straight line.
Let h, Jc he the co-ordinates of the point through which
1 the chords are drawn ; let tangents to the circle be drawn at
the extremities of one of these chords, and let (x t , y x ) be the
1 point in which they meet. The equation to the correspond-
ing chord of contact is, by Art. 100,
But this chord passes through (A, Jc) ; therefore
hx t + ky x ^c\
94 CHORD OF CONTACT.
Hence the point (x v yj lies on the line
ah + yk = <f;
that is, the locus of the intersection of the tangente is
straight line.
We will now prove the converse of this proposition*
102. If from any point in a straight line a pair oftangcA
be drawn to a circle, the chords of contact will all pass throvA
afixedpoint.
Let Ax + By + (7=0 .'(ljfl
be the equation to the straight line ; let (x' 9 jf) be a point ii
this line from which tangents are drawn to the circle ; thea
the equation to the corresponding chord of contact is
xx'+yy' = c* (2).
Since (a?', y) is on (1)
Ax' + By'+C=0;
therefore (2) may be written
i Ax'+G .
xx -y — jj — = c*,
or
{-%<-**-'- »
Now, whatever be the value of x', this line passes through
the point whose co-ordinates are found by the simultaneous
equations
*-l? = > ^+'-01
that is, the point for which
Bf _ Ac*
y_- c , x jj-.
103. The student should observe the different interpreta-
tions that can be assigned to the equation
xh+yk — c f = 0.
I&TERP&ETATIONS OP AN EQUATION,
95
I. If (h, k) be any point whatever, the equation represents
e locus of the intersection of tangents at the extremities of
oh chord through (h, Jc). (Art. 101.) . .
II. If (A, h) be an external point, the equation represents
gie chord of contact. (Art. 100.)
mf . III. If (h, k) be on the circle, the equation represents the
Jagent at that point. (Art. 91.)
In the following figures Q denotes the point (A, k), and
. jfi the line
xh+yk = (?.
In the first figure Q is within the circle, and the line RR
jceives only the interpretation I.
In the second figure Q is without the circle, hence the line
IR receives both interpretations I. and II. ; if therefore tan-
ents be drawn from Q to the circle they will meet it at the
(yints where RR intersects it.
If Q be on the circk, then RR becomes the tangent at Q.
96
CIRCLE REFERRED TO OBLIQUJB AXE*.
Oblique Axes.
^ajl (L&i* 104. To find the equation to the circle referred to a
-**&_. oblique axes.
Let © be the inclination of the axes; let C be the centrt
the circle ; P any point on its circumference. Let c be t
radius of the circle; a, b the co-ordinates of G; x,y\
co-ordinates of P. Draw ON, PM parallel to Y, and (
parallel to OX. Then
CP i =CQ 2 + PQ t -2CQ.PQcosCQP
= CQ* + PQ t +2CQ.PQcosa>;
that is, (re — a) 2 +(y — J) 2 -f 2 (sc— a) (y — b) cosa> = c*;
or, a? + y 2 + 2xy cos a> — 2 (a + J cos a>) x — 2 (5 + a cos a>) jf
+ a 2 + J 2 + 2a& cos a> — c ij
Hence the equation to the circle referred to oblique a»
of the form
a? + y* + 2ocycoaco + Aa; + By + ggQ,
where -4, B, G are constant quantities.
POLAR EQUATION TO THE CIRCLE. 97
Polar Equation.
105. To find the volar equation to the circle^ * mU*y
&T2L
CmJ
m i
8 X
Let S be the pole, BX the initial line ; C the centre of the
Sadie, JP any point on its circumference.
Let 80— I j OSX=a 9 so that Z, a are the polar co-ordi-
fetes of (7; let c he the radius of the circle; and let r, be
&6 polar co-ordinates of P.
Then CP=P8* + C8*-2P8. CB.cobPBC}
Li is, ^=r t +Z 9 -^cos(5-a) (l),
r r* — 2rZ(cosacos0 + sinasin0) + P — c* = 0.... (2).
Hence the polar equation to the circle is of the form
r* + ArcoB0 + BrBm0+C=O ^
The polar equation may also be deduced from the equation
jferred to rectangular axes in Art. 88, by putting r cos and
sin for x and y respectively.
• If the initial line be a diameter we have a = 0, hence (1)
ecomes
y*-27rcos5 + P-c 8 = (4).
If, in addition, the origin be on the circumference P = c 9 ,
.% r = 2Zcos (5).
T. C. 8. 1
I PEKPENDICULAB ON THE TANGENT.
106. To express the perpendicular from the origin at
tangent at any point in terms of the radius vector of
Let SQ be the perpendicular from the origin on the
gent at P, and suppose SQ *=p ; then
that is,
= SP 3 + PC*-2SP.PC<iosSPC
= SP' + PC t -2SP.PCainBPQ;
P=r* + c'-2cp.
In the figure Sand C are on the same side of the tan
at P. If we take P so that the tangent at P falls fc
and C, we shall find
■
the tan
a betlrt
the sol
107. These equations are sometimes useful in the w
of problems, or demonstration of properties of the circle.
example, take the equation (4) in Art. (105),
j*- 2Wcos 9 + P- c" = 0;
by the theory of quadratic equations we see that the pr
of the two values of r corresponding to any value ol
P — c*, which is independent of 9. This agrees with I
III. 35, 3G.
Also the sum of the two values of r is 2l cos 6 ; hen
line be drawn through the pole at an inclination 6 I
initial line, the polar co-ordinates of the middle point i
chord which the circle cuts off from this line are
2l cos $
Hence the polar equation to the locus
of the chord is
that is, I coa 8, and 0.
the middle
which by (5) in Art. 105, is a circle, of which the dii
is I.
EXAMPLES ON THE CIECLE. 99
EXAMPLES.
r
1. Determine the position and magnitude of the circles
(1) a? + y*+4y-4rc-l=0,
(2) x* + y*+Gx-3y-l = 0.
2. Find the points of intersection of the circle
^ + ^ = 25
rith the lines
c-
y + a; = — 1, # + # = — 5, and 3y + 4a? = — 25.
3. A circle passes through the origin and intercepts
sngths h and h respectively from the positive parts of the
xes of x and y ; determine the equation to the circle,
4. A circle passes through the points (h, h) and (h\ Tc) ;
hew that its centre must lie on the line
5. m On the line joining (x\ y) and (a?", y") as diameter a
ircle is described ; find its equation.
6. A and B axe two fixed points, and P a point such
lat AP=mBP, where m is a constant; shew that the locus
f P is a circle, except when m = l.
7. The locus of the point from which two given unequal
rcles subtend equal angles is a circle.
8. Find the equation which determines the points of
ttersection of the line
ad the circle
x*+y* - 2ax - 2by = 0.
teduce the relation that must hold in order that the line may
mch the circle.
1— *
00 EXAMPLES ON THE CIRCLE.
Find the equation to the tangent at the origin I
:
circle
a? + y*-2y — 3x
:
*
di
10. Shew that the length of the common chord o
circles whose equations are
(a; - a)* + (y - 5)' = <?, (x-b)*+(tf — a)* = c\
Vl4c'-2(a-J)'].
11. A point moves so that the sum of the squares
distances from the four sides of a square is constant ;
that the locus of the point is a circle.
12. A point moves so that the sum of the squares
distances from the sides of an equilateral triangle is cons
shew that the locus of the point is a circle.
IS. A point moves so that the sum of the aqua
distances from any given number of fixed points is cons
shew that the locus is a circle.
14. Shew what the equation to the circle becomes i
the origin is a point on the perimeter, and the axes an
clined at an angle of 120°, and the parts of them intern
by the circle are h and k.
15. What must be the inclination of the axes thai
equation
x t + y t — xy —Jtx — liy =
may represent a circle ? Determine the position and m;
tude of the circle.
16. What must be the inclination of the axes tha
equation
x 1 + y* + xy — fix — hy =
may represent a circle? Determine the position and m:
tude of the circle.
17. Determine the equation to the circle which ha
centre at the origin, and its radius = 3, the axes beinj
clined at an angle of 45°.
EXAMPLES ON THE CIRCLE. 101
18. Determine the equation to the circle which haa each
2
f the co-ordinates of its centre = — i and its radius = -7- ,
he axes being inclined at an angle of 60°.
19. The axes being inclined at an angle <o, find the radius
f the circle
a? + y* + 2xy cos a> -<■ hx — ley = 0.
20. Shew that the equation to a circle of radius c referred
> two tangents inclined at an angle <o as axes is
# , +y 2 + 2a3ycosa>~ 2 (tf+^ccot-H-^cot 8 - = 0.
21. Shew that the equation in the preceding question
lay also be written
K + y- 2 \/(xy) sin| = c cot |.
22. Find the value of c in order that the circles
(x-a)*+(y-b)*~c*, and (x - J) 9 + (y - a) 2 » c 2 ,
iay touch each other.
23. -4J5(7 is an equilateral triangle ; take A as origin,
nd AS as axis of x ; find the rectangular equation to the
irele which passes through -4, B, G. Deduce the polar equat-
ion to this circle.
24. If the centre of a circle be the pole, shew that the
olar equation to the chord of the circle which subtends an
ngle 2p at the centre is
r = c cos 13 sec (0 — a),
'here a is the angle between the initial line and the line from
le centre which bisects the chord. Deduce the polar equa-
on to a line touching the circle at a given point.
25. Find the polar equation to the circle, the origin being
n the circumference and the initial line a tangent. Skew
102 EXAMPLES ON THE CIRCLE.
that with this origin and initial line, the polar eqi
tangent at the point ff is
I
r Bin (20' -0) = 2c Bio* 0*,
26. Shew that if the origin be on the circumferer
a circle, and the diameter through that point make an an
with the initial line, the equation to the circle is
r = 2c eoa(0 -a).
27. Determine the locus of the equation
r = ^cos (0-a) + Bcos(0-£) +C cos (0- 7 )+.„.
28. AB i3 a given straight line ; through A two
finite straight lines are drawn equally inclined to AB
any circle passing through A and B meets those lin
L, M; shew that the sum of AL and AM is equal
constant quantity when L and M are on opposite sides ol
and that the difference of AL and AM 13 constant wl
and J/ are on the same side of AB.
29. -^.BC is an equilateral triangle, and
PA = PB + PO,
find the locus of P.
30. There are n given straight lines making with ai
fixed straight line angles a, ft, y, ; a point P is
such that the sum of the squares of the perpendicular
it on these n lines is constant; find the conditions tin
locus of P may be a circle.
31. A point moves so that the sum of the squares
distances from the sides of a regular pol;
shew that the locus of the point is a circle.
32. A line moves so that the sum of the perpendi
AP, BQ, from the fixed points A and B is constant
the locus of the middle point of PQ.
33. is a fixed point and AB a fixed line ; a I
drawn from meeting AB in P; in OP a point Q ig
so that OP.OQ = tf; find the locus of Q.
dra
so
EXAMPLES ON THE CIRCLE. 103
84. A line is drawn from a fixed point 0, meeting a fixed
fade in P; in OP a point Q is taken so that OP. OQ=l?\
aid the locus of Q.
©presents the two tangents to the circle,
af + ^c",
rhich pass through the, point (A, Jc).
36. What is represented by the equation
t*«-ra cos 20 sec - 2a 2 = 0?
'[ 37. The polar equation to a circle being r = 2c cos 0, shew
hat the equation
2c cos ft cosa = r cos 08 + a — 0)
epresents a chord such that the radii drawn to its extremities
rom the pole, make angles a, ft with the initial line.
38. Tangents to a circle at the points P and Q intersect
tk T\ if the lines joining these points with the extremity
rf a diameter cut a second diameter perpendicular to the
fanner in the points p, q, t, respectively, shew that
pt = qt.
p
i
CHAPTER VII.
RADICAL AXIS. POLE AND POLAR.
Radical Axi$.
108. We have shewn that the equation to a circle is
(a?-a)*+(y-5) , -o , *-0.
We shall write this for abbreviation
8=0.
If the point (x, y) be not on the circumference of thec&A
8 is not = ; we may in that case give a simple geometop
meaning to 8.
I. Let (z, y) be without the circle ; draw a tangent fa*
(x, y) to the circle ; join the point of contact with the cflW
of the circle (a, ft) ; also join (x, y) with (a, ft). Let ft
present the point (a, ft), Q the point (x,y) f and !T the p^*
of contact of the tangent. Thus we have a right-angw
triangle formed, and since (x — a) 2 + {y — ft) 2 = QG f , it folfo*
that 8= QT % ; that is, 8 expresses the square of the tang^
from (x,y) to the circle. By Euclid in. 36, the square of 4?
tangent is equal to the rectangle of the segments made by fc
circle on any straight line drawn from (a?, y), and thus b U»
also express the value of this rectangle.
II. Let (x,y) be within the circle; then 8 is negative
Let G and Q have the same meaning as before, and proto
CQ to meet the circle in T and T'\ then
- A* CT*- C(? = (GT- CQ) {CT+ CQ)
** =TQ.T'Q.
RADICAL AXIS. 105
nee by Euclid III. 35, if any line PQP' be drawn meet-
i circle in P and P', the value of the rectangle PQ . F Q
I, Let 8 denote (x — a)" + (y — b) % — c 1 ,
S' denote (x - o')*+ (y - £')*— c";
5=0 (1), and S' = (2),
: equations to two circles ; we proceed to interpret the
>n
8-S' = Q (3).
contains only the first powers of x andy; therefore
= is the equation to some, straight line. Also it'
of a; and^ can be found to satisfy simultaneously (1)
!), these values will satisfy (3). Hence when the
represented by (1) and (2) intersect, (3) is the equa-
) the straight line which joins their points of inter-
na suppose that from any point in (3), external to both
we draw tangents to (1) and (2) ; then, by Art. 108,
angents are equal in length. Hence whether (1) and
srsect or not, the line (3) has the following property; —
i any point of it lines be drawn, to touch both circles, the
of these lines are equal.
). An equation of the form
A (a? + y*) + Bx + Gy + D =
ipresent a circle ; for after division by A we obtain the
•j form of the equation to a circle. We shall say that
lation to a circle is in its simplest form when the co-
t of iB* and j* i3 unity.
F. If S= 0, S' = 0, be the equations to two circles in
implest forms, the straight line S-S' = is called the
I axis of the circles,
e axes of co-ordinates may here be rectangular or oblique.
we may give a geometrical definition thus. A straight
,n always be found such that if from any point of it
ts be drawn to two given circles, these tangents are
tbia line i3 called the radical axis of the circles.
106
RADICAL AXIS.
111. The three radical axes belonging to three given
meet in a point.
Let the equations to the three circles be
$-0 (1), £, = (2), S s = (3).
The equations to the radical axes are
S x - S t = 0, belonging to (1) and (2),
# 2 -fl, = 0, (2) and (3),
£ f -£ i= =o, (3)and(l).
These three lines meet in a point ; since it is obvious
the values of x and y which simultaneously satisfy two of
equations, will also satisfy the third.
112. A large number of inferences may be drawn
the preceding articles by examining the special cases
fall under the general propositions. (See Pllicker Anai^
Geometrische MntwicJcelungen, Vol. I. pp. 49 — 69.) We
a few of these respecting the radical axis of two circles.
113. The radical axis is perpendicular to the lineji
the centres of the two circles.
Let the equations to the circles be
(a;-a)»+(y-i)»-c 2 = 0,
then the equation to the radical axis is
( a ,-a) i -(»-a # ) i +(y-J) i -(y-6 # ) i -J , + rf i «0;
that is,
x(a'-a)+y(V-b)+i{a*-a , *+l*-b*-c*+c^=0 fl
And the equation to the line joining the centres rf
circles is (Art. 35)
y-" & =^— -(«-<*)
a —a
(1) and (2) are at right angles by Art. 42.
(»)".;
RADICAL AXIS.
107
.14. When two circles touch, their radical axis in the
imon tangent at the point of contact. For the radical axis
tea through the common point and is perpendicular to the
joining the centres of the circles.
.15. Suppose the radius of one of the circles to become
! Snitely small, that is, the circle to become a point ; the
cal axis then ha3 the following property: — it from any
it of the radical axis we draw a line to the given point,
a tangent to the given circle, the line and the tangent
be equal in length.
116. The radical axis of apoint and a circle falls without
circle, whether the point be without or within the circle.
if the radical axis met the circle, the co-ordinates of the
its of intersection would satisfy the equation to the point as
L as the equation to the circle. But the equation to the
it can be satisfied by no co-ordinates except the co-ordi-
a of that point ; therefore the radical axis cannot meet
circle. If the point be on the circle, the radical axis is
tangent to the circle at this point.
117. Suppose loth eircles to become points. Then the
a drawn from any point in the radical axis to the two
d points are equal in length. Hence the radical axis be-
?ing to two given points is the line which bisects at right
les the distance between the two given points.
118. Suppose in Art, 111 that each circle becomes a point;
theorem proved ia then the following : — the perpendiculars
vn from the middle points of the sides of a triangle meet
point.
119. It is a well-known geometrical problem— to draw a
ight line which shall touch two given circles. If the circles
lot intersect, four common tangents'can be drawn; two of
a will be equally inclined to the line joining the centres,
will intersect on that line between the circles ; the other
will also be equally inclined to the line joining the cen-
, and will intersect on that line beyond the smaller circle.
i two points of intersection are called centres of similitude.
^ two poin
108 POLE AND POLAE.
For the equations to the common tangents and for the pro
ties of the centres of similitude, we refer to Salmon's (■•
Sections.
Pole and Polar.
120. Def, If the equation to a circle be
*■+/-■=",
and h, h be the co-ordinates of any point, then the line
xh + yk = c*
is called the polar of the point (A, k) with respect to the gn
circle, and the point [h, k) is called the pole of the line
xh + yk = c*
with respect to the given circle.
We may also express our definition thus: — the polar of
given point with respect to a given circle is the straight 6
whose equation involves the co-ordinates of the given pi
in the same manner as the equation to the tangent at<
point of the circle involves the co-ordinates of the point
contact ; and the given point is the pole of the line.
This definition might he misunderstood. For the ep
tion to the tangent to a circle at a given point might
expressed in different forms by using the relation which U
between the co-ordinates of the given point by virtue of I
equation to the circle. We might for example express I
equation to the tangent in terms of either of the co-ordini
of the given point alone. But in the above definition we on
that the equation to the tangent is to be in the form whid
naturally assumes, involving the co-ordinates of the gT
point rationally.
Or we may define the polar of a point by means of
properties which it possesses (Art. 103). The polar oi
given point with respect to a given circle is the straight
which is the locus of the intersection of tangents drafft
the extremities of every chord through the given point;
the given point is called the pole of this straight line.
If the given point he without the circle, its polar coinc
with the chord of contact of tangents drawn from that
-in that paH
POLE AND POLAE. 109
1. If one straight line pass through the pole of another
it line, the second straight line will pass through the pole
first straight line.
t (x\ y) be the pole of the first straight line, and
>re
xx' + yy' = <? (1)
uation to the first straight line.
>t (a?", y") be the pole of the second straight line, and
Dre
xx" + yy" = <? (2)
nation to the second straight line.
nee (1) passes through (x", y") we have
x"x' + y"y' = <?;
ince this equation holds, (2) passes through (a/, y f ).
52. The intersection of two straight lines is the pole of the
^hich joins the poles of those lines.
enote the two straight lines by A and B, and the line
ig their poles by G\ since G passes through the pole of
lerefore, by Art. 121, A passes through the pole of G;
urly B passes through the pole of G\ therefore the inter-
in of A and B is the pole of 0.
MISCELLANEOUS EXAMPLES.
» Find the tangent of the angle between the two straight
whose intercepts on the axes are respectively a, J, and
• If the straight lines represented by the equation
a?(tan f £+ cos 2 ^) - 2ajytan£ + y*sin*£=*0,
) angles a, fi with the axis of x, shew that
tan a ~ tan /8 =» 2.
110 MISCELLANEOUS EXAMPLES.
3. One aide of a square a corner of which is at the or
makes an angle a with the axis of x ; find the equation
the four sides and the two diagonals.
4. Find the equations to the diagonals of the parallelog
formed by the straight lines
r+f-».
JL =
h r+« = 2;
ther.
and shew that they are at right angles to one anoi
5. The distance of apoint (x lt yj from each of two st«i|
lines which pass through the origin of co-ordinates is Sj port
that the two lines are represented by the equation
(x [ t/-xy 1 )'=(x t +f)^.
6. Find the condition that one of the lines represented
Ay* 4- Bxy + CV =
may coincide with one of those represented by
ay* + bxy + ex* =0.
1. If a = 0, £ = 0, 7 = he the equations to the &
sides of a triangle ; and a, b, c he the perpendicular distal)
lietween these sides and those of another triangle parallel
them respectively, the line joining the centres of the inscri'
circles will be represented by any of the equations
«-0 ,S- T _ T -«
8. Prove that the equation to the straight line pass;
through the middle point of the side BO of a triangle A,
and parallel to the external bisector of the angle A is
B + y+~{smB + S inC) = 0.
MISCELLANEOUS EXAMPLES. Ill
9. The equation to the line drawn parallel to BG through
le centre of the escribed circle which touches BG is
{a+P) sin B+ (a + 7)sin (7=0.
10. Find the equations to the lines which pass through
ie intersection of the lines
fee + w/S+ ny = 0, l'a + m'p + n'y = 0,
id "bisect the angles between them.
11. If u = 0, v = 0, be the equations to two circles, shew
tat by giving a suitable value to the constant \, the equation
-+ \v = will represent any circle passing through the points
? intersection of the given circles.
»
12. A fixed circle is cut by a series of circles, all of which
Ms through two given points ; shew that the lines which
wn the points of intersection of the fixed circle with each
bele of the series all meet in a point.
( 112 )
CHAPTER VIII.
THE PAEABOLA.
123. There are three curves which we now procet
define ; we shall then deduce their equations from the d<
tions, and investigate some of their properties from
equations.
Def. A conic section is the locus of a point which in
d that its distance from a fixed point bears a constant rat
its distance from a fixed straight line. If this ratio be h
the curve is called a parabola , if less than unity, an ellj£
greater than unity, an hyperbola .
The fixed point is called the focus, and the fixed strs
line the di rectrix .
124. It will be shewn hereafter that if a cone be en
a plane, the curve of intersection will be one of the follow
ii parabola, an ellipse, an hyperbola, a circle, two stn
lines, one straight line, or a point. Hence the term i
section is applied to the parabola, ellipse, and hyperbola-
may be extended to include the circle, two straight lines,
straight line and point. We shall also prove that e
curve of the second degree must be a conic section in
larger sense of the term.
At present we confine ourselves to tracing the coi
of the definitions in Art. 123.
tl25. To find the
A parabola is the
distance from a fixed
fixed straight line.
nseqne
A parabola is the locus of a point which moves so ths
distance from a fixed point is equal to its distance fro
fixed straight line.
EQUATION TO THE PARABOLA.
113
8 be the fixed point, YY' the fixed straight line.
W perpendicular to YY'; take as the origin, OS
lirection of the axis of a?, OF as that of the axis of y.
>OS=2a.
P be any point on the locus; join SP; draw PM
to OF and PN parallel to OX; let OM=x,
lefinition
SP=PN;
.-. SP* = PN*;
.: PM* + SM* = PN*,
y 2 + (a-2a) 2 = ^;
/. y 2 = 4a(a; — a)
a).
ss is the equation to the parabola with the assumed
,nd axes. The curve cuts the axis of a? at a point A
jisects 08; for when y = Q in (1), we have x = a.
nation will be simplified if we put the origin at A ;
AM, then x' = x — a, and (1) becomes
y 8 = Aax'.
•>• o*
%
114 FOEM OF THE PARABOLA.
We may suppress the accent, if we remember tit
origin is now at A ; thus we have for the equation
parabola
v'^iax
126. To trace the parabola from its equation y* = ii
From this equation we see that for every positive
of x there are two values of y, equal in magnitude,
opposite sign. Hence for every point P on one side
axis of x, there is a point P' on the other side, su<
P'M=PM. Hence the curve is symmetrical with
to the axis of x. Negative values of x do not give t
values of y; hence no part of the curve lies to the lefl
origin. As x may have any positive value, the curve t
without limit on the right of the origin.
Ah
curve.
called the vertex of the curve and AX the axis
127. We have drawn the curve concave towards t
of a;; the following proposition will justify the figure.
The ordinate of any point of the cwve which lies h
the vertex and a fixed point of the curve is greater tl
corresponding ordinate of the straight line joining the
and the fixed point.
FOCAL DISTANCE OF ANY POINT. 115
Let P be the fixed point; x\ y' its co-ordinates; then the
lation to AP is
se y t% = 4aa/.
Ijet x denote any abscissa less than x\ then since the ordi-
e of the curve is *J(±ax), and that of the straight line is
(—,).x or a/(— r) X\/(4aa;), it is obvious that the ordi-
e of the curve is greater than that of the line.
128. Def. The double ordinate through the focus of
ftnic section is called the Latus Rectum.
Thus in the figure in Art. 126, LSL' is the Latus Rectum.
Let a? = a, then from the equation y 2 =4oaj, y=±2a.
Race LS=L'S=2a ; and LU = 4a.
129. To express the focal distance of any point of the f
rabola in terms of the abscissa of the point.
lie distance of any point on the curve from the focus is
nal to the distance of the same point from the directrix.
mce (see fig. to Art. 125),
SP=AM+AS,
= x + a.
Tangent and normal to a Parabola.
130. To find the e quation to the tangent at any point of
parabola. (See Def. Art 90.)
Let x\ y be the co-ordinates of the point,
a*", y" the co-ordinates of an adjacent point on the
curve.
116 TANGENT TO A PAEABOLA.
The equation to the secant through these points is
*-*'-£r£<"-* f > (
since (x', y) and (x", y") are on the parabola
y' J =4aa?', y"* = ±ax":
-' x"-x'-y" + y"
hence (1) may be written
• r 4a , , N
Now in the limit y" = y'; hence the equation to the ta
at the point (x\ y') is
y-y =y (*-*)
This equation may be simplified ; multiply by y\ tl
yy' = 2a{x-x')+y'* >
= 2ax — 2ax' + ±ax\
= 2a(x + x')
131. The equation to the tangent can be conve:
expressed in terms of the tangent of the angle which t
makes with the axis of the parabola.
For the equation to the tangent at {x\ y') is
yy' = 2a(x+x) 9
2a 2ax'
or y = — x H -
J y y
__ 2a Aax
2a v'
= -7«+%
y 2
TANGENT TO A PARABOLA. 117
2a
—. =m:
• £=^.
y' 2 m
.) may be written
y = mx + — ... (2);
m
the required equation. Conversely, every line whose
>n is of this form is a tangent to the parabola.
J. It may be shewn as in Art. 93, that a tangent to
rabola meets it in only one point. Also, if a line meets
bola in only one point, it will in general be the tangent
; point.
r suppose
y*=4aa? (1)
he equation to a parabola, and
y = mx + c (2)
lation to a straight line. To determine the abscissae of
ints of intersection, we have the equation
(mx + c)* =■ 4oas,
wV + (2mc-4a)a? + c 2 =0 (3);
ladratic equation will have two roots, except when
(wc — 2a) 2 = mV,
, when
a
m
>nce if the line (2) meets the parabola, it will meet it in
>ints, unless c = — , and then the line is a tangent to
m
rabola by Art. 131.
however, the equation (2) be of the form y = c, so that
e is parallel to the axis of x, then instead of (3) we have
nation c* = 4o#, which has but one root ; hence a line
118 NORMAL TO A PARABOLA.
parallel to the axis of the parabola meets it in only one point
but is not a tangent.
133. The axis of y is a tangent to the curve i
vertex.
For the equation to the tangent at (2:', y') is
yy' = 2a (x+x')- t
and when x' = and y = 0, this becomes
\, vmal 134. To find the equation to the n ormal at any point'
SS ^ a parabola. (See Def* Art. 97.)
Let x',y' be the co-ordinate3 of the point; the equation
to the tangent at that point is
y = y{x+x') (1).
The equation to a line through (x', y') perpendicular to
(1)13
y-/--£(*-«0 (2).
This is the equation to the normal at (x, y).
135. The equation to the normal may also be expressed
in terms of the tangent of the angle which the ,: ~
with the axis of the curve.
For the equation to the normal is
y =
y
2a
*+»'+§
or
F"
y 1
2a
* + "' + £
Let
y' _
2a
-;
■■■ y--
2am;
tli
1.(1)
may
be written
y=
mx -
2am — am".
PROPERTIES OF THE PARABOLA.
119
136. We shall now deduce some properties of the parabola f^
om the preceding articles.
Let x' 9 y' be the co-ordinates of P; let PT be the tangent
l JP and PG the normal at P.
The equation to the tangent at P is
yy' = 2a(aj+aj').
Lety = 0, then a? = — x'; hence AT*=*AM.
Also BT-AT+A8,
=AM+AS,
= SP (Art. 129).
Hence the triangle 8TP is isosceles, and the angle 8TP
= angle SPT.
Thus if PN be parallel to the axis of the curve, PN and
*S are equally inclined to the tangent at P.
137. The equation to the normal at Pis
120 PROPERTIES OF THE PARABOLA.
At the point O, where the normal cats the axis* jr=t
hence from the above equation
x — x' = 2a ;
thus MO = 2a = half the latus rectum. Also SG = BP.
138. To find the locus of the intersection of the tanged A
any point with the perpendicular on it from the focus.
Let x\ y be the co-ordinates of any point P on the cum;]
the equation to the tangent at P is
y=y (» + a0 (1).
The equation to a line through the focus perpendu
to (1) is
*--£<"-•>••< «■
We have now to eliminate x and y' by means of (1).
(2), and
y'*=±ax' (3).
From (3) we find x in terms of y\ and thus (1) may k
written
2« V
3^ = 7^ + 1 M-
Thus the problem is reduced to the elimination of y' fioo
(2) and (4) ; from (2)
/--£ • »
substitute in (4) ; then
__ (x—a) x ay
'V y x — a 9
•'• y"(a> — a) + (o> — a) J o> + ay 2 = 0,
or {y*+ (aj-a) J }a=:0 (6).
If the factor y f + {x — a) 2 be equated to zero, we have
y = 0, x = a (7).
LOCUS OBTAINED BY ELIMINATION. 121
The point thus determined is the focus ; this however is
feurt the locus of the intersection of (1) and (2), for the values
moi (7), although they satisfy (2), do not satisfy (1). We
conclude therefore that the required locus is given by the
equation
* fc (8),
^hich we obtain by considering the other factor in (6).
This result can be easily verified ; for if we put x = in
" 1) we obtain y = — — = ^ ; and if we put x = in (2), we
y 2
also obtain y = ^- ; thus (1) and (2) intersect on the line
■* = 0.
Thus, if in the fig. in Art. 136, Z be the intersection of the
aangent at Pwith the axis of y, SZ is perpendicular to the
tangent.
139. The process of the preceding article is of frequent
use and of great importance. We have in (1) and (2) the
equations to two straight lines ; if we obtain the values of x
■nd y from these simultaneous equations, we thus determine
the point of intersection of the lines ; the values of x and y
will depend upon those of oi and y\ thus giving different
points of intersection corresponding to the different lines re-
presented by (I) and (2). If from (1), (2), and (3) we elimi-
nate x and y we obtain an equation which holds for the
co-ordinates of every point of intersection of (1) and (2). This
equation is by our definition .of a locus the equation corre-
sponding to the locus of the intersection of (1) and (2).
Sometimes the elimination produces, as in the preceding
article, an equation which does not represent the required
Locus. The student has probably noticed in solving alge-
braical questions that he often arrives at more results than
that which he is especially seeking. We can frequently
interpret these additional results; thus in the preceding
article, since, whatever x' and y' may be, the values x = a,
y = 0, satisfy one of the equations which we use in effecting
the elimination, we might anticipate that our result would
involve a corresponding factor.
\
122 PERPENDICULAR ON THE TANGENT.
140. If the line from the focus, instead of being per
dicular to the tangent, meet it at any constant angle,
locos of their intersection will still be a straight line,
will indicate the steps of the investigation. Suppose j3
angle between the tangent and the line from the fo
equation (1) remains as in Art. 138 ; instead of (2) we 1
by Art. 45,
2« x Q
-7-|-tanj8
y = ir (x — a)
1 ; tan p
y
= 2ojyQan|
y-2atan£ v '
Instead of (5) in Art. 138, we shall find
, _ 2a (x — a) + 2ay tan ft
* "" y — (# — a) tan£
The result of the elimination is
y [y "" ( x ~~ a ) * an &} \ x — a + y * an &}
— x[y — (a? — a) tan/8} 8 — a (a? — a + y tanj8
Now, guided by the result of Art. 138, we may anti
that y* + (x —a) 2 will prove a factor of the left-hand m
of the equation ; and we shall find by reduction that the
tion may be written
{y*+ (#-0)*} [y tan/8 -a? tan'£-a} = 0.
Hence the required locus is
y = a? tan # + a cot #.
141 . 2b J? nc? the length of the perpendicular from th
on the tangent at any point of the parabola.
The equation to the tangent at the point (a?', y') is
2a . , v
TWO TANGENTS FROM AN EXTERNAL POINT. 123
The perpendicular on this from the point (a, 0) by Art. 47
2a(a + x') 2a(a + x') ., , , , u
Call the focal distance of the point of contact r, and the
erpendicular p ; then, by Art. 129,
r = a + x;
142. From any external point two tangents can he drawn 'J**** "**
» a yarabola. \ZZjt
Let the equation to the parabola be
y* = 4oa; (1),
lid let A, h be the co-ordinates of an external point. Sup-
ose x, y 1 the co-ordinates of a point on the parabola such
hat the tangent at this point passes through (A, h). The
quation to the tangent at (x, y') is
yy* = 2a (a? + x') (2).
Since this tangent passes through (A, h)
ky'=*2a(h + x') (3).
Also since (x, y) is on the parabola
y"=4aa/ (4).
Equations (3) and (4) determine the values of x' and y\
Substitute from (4) in (3), thus
Ay'=2aA+^-,
p y * - 2hy' + lah = 0.
The roots of this quadratic will be found to be both pos-
i_ ,t . -, • , srefore A* greater
one value of x'
Me, since (A, k) is an external point and therefore A* greater
lan 4aA. To each value of y corresponds
124 ■ CHORD OF CONTACT.
by (3) ; beuce two tangents can be drawn from any extei
point.
The line which passes through the points where these I
gents meet the parabola is called the chord of contact.
143. Tangents are drawn to a parabola from a gin
external point ; to find the cgaaiiun to thy„<J\ord of contact .
Let h, h be the co-ordinates of the external point; zj
the co-ordinates of the point where one of the tangents fra
(h, k) meets the parabola ; as,, y 2 the co-ordinates of the poii
where the Other tangent from (h, k) meets the parabola.
The equation to the tangent at (x l , y,) is
yy, = 2a{x + x 1 ) (l),
Since this tangent passes through (h, k) we have
ky l =2a{h + x 1 ) (!).
Similarly, since the tangent at (at,, yj passes throng
(M)
ky, = 2a(h + x 1 ) (3).
Hence it follows that the equation to the chord of cm
tact is
ky = 2a{x+h) (4).
For (4) is obviously the equation to some straight line ; li
this line passes through (x lt y^j, for (4) is satisfied by tif
values x = x„ y =y , as we see from (2); similarly from
we conclude that this line passes through (at,, yj. Hentt I
is the required equation.
Thus we may proceed a3 follows in order to draw tangn"
to a parabola from a given external point. Draw the 1
which is represented by (4), join the points where it am
the parabola with the given external point, and the lines tbu. 1
obtained are the required tangents.
144. Through any fixed point chords are drawn to t
bola, and tangents to the parabola drawn at the exit-
each chord; — the locus of the intersection of the tangents o*
straight line.
CHOBD OF CONTACT. 125
Let h, h be the co-ordinates of the point through which the
ords are drawn; let tangents to the parabola De drawn at
e extremities of one of these chords, and let (x l9 y x ) be the
>int in which they meet. The equation to the corresponding
lord of contact is, by Art. 143,
yy 1 = 2a(x + x l ).
But this chord passes through (A, k) ; therefore
iy x = 2a (A+a? x ).
Hence the point (x l9 yj lies on the. line
Icy = 2a (x + h) ;
iat is, the locus of the intersection of the tangents is a
raight line.
We will now prove the converse of this proposition.
145. If from any point in a straight line a pair of tangents l
j drawn to a parabola, the chords of contact will all pass
trough a fixed point
Let Ax + By + (7=0 (1)
e the equation to the straight line ; let (x', y') be a point in
his line from which tangents are drawn to the parabola;
lien the equation to the corresponding chord of contact is
yy' = 2a {x + x') (2).
Since (x\ y') is on (1)
Ax' + By'+C=0;
herefore (2) may be written
y (Ax' +C) + 2aB (x + x') = 0,
>r (Ay + 2aB)x' + Cy + 2aBx = (3).
Sow whatever be the value of x', this line passes through
lie point whose co-ordinates are found by the simultaneous
jquations
Ay + 2aB = 0, Cy+2aBx=0;
V
126 INTERPRETATIONS OF AN EQUATION.
that is the point for which
2aB G
146. The student should observe the different interpiA
tions that can be assigned to the equation
ky = 2a (x + A).
The statements in Art. 103 with respect to the circle m»yi
be applied to the parabola. | *A
Diameters. J*- - ,"
i
—^
147. To find the length of a line drawn jr 6m any points
a given direction to meet a parabola.
Let x\ y be the co-ordinates of the point from which
line is drawn ; a?, v the co-ordinates of the point to which
line is drawn ; the inclination of the line to the axis of*l
r the length of the line ; then (Art. 27)
x = x + r cos 0, y =y r + r sin (1).
If (x, y) be on the parabola, these values may be substitute!
in the equation y 2 = lax; thus
■
{y + r sin 0)* = 4a (x' + r cos 0) ;
.*. r*sin*0 + 2r(y'sin0-2acos0) + y v> -4ag'«0...(3)-
From this quadratic two values of r can be found, wMA
are the lengths of the lines that can be drawn from (a/jjl
in the given direction to the parabola.
When the point (x\ y') is within the parabola, the roots i
the above quadratic will be of different signs ; in this ca*
the two lines that can be drawn from (x\ y) to meet fc
curve are drawn in different directions. When the po^
(x\ y) is without the parabola, the roots are of the same sig«i
and the lines are drawn in the same direction.
DIAMETER. 127
148. Dep. A diameter of a curve is the locus of the
idle points of a series of parallel chords.
149. To find the diameter of a given system of parallel
yards in a parabola.
Let be the inclination of the chords to the axis of the
rabola ; let x'> y' be the co-ordinates of the middle point
any one of the chords ; the equation which determines the
igths of the lines drawn from (x' f y') to the curve is
XL 147)
r*8w* + 2r (y' airiO - 2acos0) +y'* - lax' = (1).
Since {x\ y') is the middle point of the chord, the values
r furnished by this quadratic must be equal in magnitude
id opposite in sign; hence the coefficient of r must vanish;
ins
y' sin — 2a cos = ;
•\ y' = 2acot0 (2);
ins the required diameter is a straight line parallel to the
ris of the parabola.
Hence every diameter is parallel to the axis of the para-
>la.
Also every straight line parallel to the axis of the para-
>la is a diameter, that is, bisects some system of parallel
lords ; for by giving to a suitable value, the equation (2)
iay be made to represent any line parallel to the axis.
150. Let a tangent be drawn to the parabola at the point
here the line y = 2a cot 9 meets the parabola; the equation
> the tangent is
y = y (*+*);
iat is, y = tan {x + x') ;
moey the tangent at the extremity of any diameter of the
orabola is parallel to the chords which that diameter bi-
Kts.
128 EQUATION TO THE PARABOLA REFERRED
v- 151. To find the equation to the parabola, the axes bm
any diameter and the tangent at the point where it wt
the curve.
JJet h, h be the co-ordinates of a point A 1 on the paiabc
take this point for a new origin ; draw through it a 1
A'X' parallel to the axis of the curve for the new axis
x, and a tangent A' Y' to the curve for the new axis oJ
Let TA'X'^0; then (Art. 150)
j- = tan 0.
Let x, y be the co-ordinates of a point P on the cb
referred to the original axes ; x\ y' the co-ordinates of
same point referred to the new axes ; draw PM paralU
A Y and PM 1 parallel to A' Y' ; also draw A'L, wNjM
to AY; let B denote the intersection of PM and A'X^**
<"~4M=AL + LN+NM=AL + A'M' + M
i+a?' + y'cos0,
TO A DIAMETEB AND TANGENT AS AXES. 129
y = PM=EM + P£ = A'L + P£
Substitute these values in the equation y* = 4ax ; thus
(& + y' sin 0) 8 = 4a (h + x +y' cos 0),
y'*am'0+ 2y' (&sin - 2a cos 0) + A* - 4aA = 4ax\
But, A = 2a cot ft and &*= 4aA ; thus we have
y 2 sin* = 4oa>',
4a
sm*0
ich is the required equation.
We may prove that
V ~~ ZJZXa x »
sin
SA' = a + h (Art. 129) ; and
sur0
Hence the equation may be written
y ,8 =4aV,
ere a' = SA \; or suppressing the accents on the variables
y % = 4a'sc.
152. The equation to- the tangent to the parabola will be
the same form whether the axes be rectangular, or the
ique system formed by a diameter and the tangent at its
remity ; for the investigation of Art. 130 will apply with-
any change to the equation y* = Aax which represents a
abola referred to such an oblique system.
1 53. Tangents at the extremities of any chord of a parabola
t in the diameter which bisects that chord.
T. c. s. 9
130
POLAR EQUATION,
Refer the parabola to the diameter bisecting the chord,!
the corresponding tangent, as axes ; let the equation to
parabola be
y 2 = 4a'sc ;
let x', y' be the co-ordinates of one extremity of the cho
then the equation to the tangent at this point is
yy' = 2a'{x + x') (1).
The co-ordinates of the other extremity of the chord
x\ — y ; and the equation to the tangent there is
-tf = M{x + af) (2).
The lines represented by (1) and (2) meet at the point!
which
this proves the theorem.
x = — x ;
Polar Equation.
154. To find the Tfalar Equation in tJ»A jru^^nln^ (k d
being the pole.
Let SP=r, ASP=6, (see Fig. to Art. 125) ;
then SP = PN> by definition ;
that is, SP = OS+SM;
or r = 2a + r cos {ir — 0) ;
.'. r (1 + cos 0) = 2a,
2a
and
r =
1 + cos '
If we denote the angle X8P by 0, then we have a»
SP=0S+8M;
thus r = 2a + rcos0,
2a
and
r =
1 — cos '
FOCAL CHORDS, 131
j 155. The polar equation to the parabola when the vertex
-the pole may be conveniently deduced from the equation
*= 4oa? by putting r cos and r sin for x and y respeo-
rely j we thus obtain
4a cos
r ~ sin*0 '
"We add a few miscellaneous propositions on the parabola.
Def. A chord passing through the focus of a conic sec-
a is called a focal chord.
156. If tangents be drawn at the extremities of any focal
*rd of a parabola, (1) the tangents will intersect in the
ectrtXy (2) the tangents will meet at right angles, (3) the
e drawn from the point of intersection of the tangents to
Jbcus will be perpendicular to the focal chord.
(1) If the tangents to a parabola meet in the point (h 9 k)
fe equation to the chord of contact is, by Art. 143,
Jcy = 2a (x + h).
Suppose the chord passes through the focus; then the
lines x — a, y *±0, must satisfy this equation ;
•\ A = — a;
iat is, the point of intersection of the tatigfents is on the
irectrix.
(2) The equation to the tangent to a parabola may be
ritten (Art. 131)
y = mx H — .
m
Suppose (A, h) a point on the tangent ;
.'. hm 2 — Jcm + a = Q.
This quadratic will determine the inclinations to the axis
F the parabola of the two lines that may be drawn through
132 EECTANGLE OF THE SEGMENTS OF A LINE.
the point (A, k) to touch the parabola. Suppose i»,, m,
tangents of these inclinations, then by the theory of <]i
ratio equations
If A = — a, mjJMj = — 1 ;
that is, the two tangents are at right angles.
(3) The equation
(A, Jt) is
the line through the focus
j—— {x-a}.
and the line is therefore perpendicular to the focal choi
which the equation is
yk = 2a{x-a).
157. If through any point within or without a para
two lines be drawn parallel to two given straight lines to
the curve, the rectangles of the segments will be to one am
in an invariable ratio.
Let {x', y) be the given point, and suppose a an
respectively the inclinations of the given straight line
the axis of the parabola. By Art. 147, if a line be di
through (x, y') to meet the curve and be incbned at an a
a to the axis, the lengths of it3 segments are given by
equation
r* sin* a. + 2r {y' sin a — 2a cos a) + y" — iax' = 0.
Therefore by the theory of quadratic equations the i
angle of the segments
y' 1 — iax'
RECTANGLE OP THE SEGMENTS OP A LINE.
133
Similarly the rectangle of the segments of the line drawn
ugh (x', y') at an angle /3
_ y'*-±ax'
~ sin'ft #
Hence the ratio of the rectangles = . , ,
° sin 8 a
fail this ratio is constant whatever x' and %/ may be.
m
i
Let 0be the point through which the lines OPp, OQq,
SB drawn inclined to the axis of the parabola at angles a, p,
jfepectively ; then we have proved that
$ OP. Op sin 8 ft,
I OQ.Oq sin'V
Let tangents to the parabola be drawn parallel to Pp, Qq,
eting the parabola in E and D respectively ; let 8 be the
*eus; then by Art. 151,
flff sin'ft . OP. Op SE
£Z>~*sin*a ; •"• 0Q.0q~8D'
Suppose to coincide with T\ then OP. Op becomes
"3E" and Q . Oq becomes 7LD 8 ;
TE* SE
•"• TJD*~ SD'
EXAMPLES OS THE PAKABOLA.
1. Find the equation to the line joining A a
Fig. to Art. 126.)
2. Find the equation to the circle which passes tl
A, L, L: (See Fig. to Art. 126.)
3. A point moves so that its shortest distance fr<
given circle is equal to its distance from a given fixed
meter of that circle ; find the locus of the point.
4. Trace the curves if = iax, and x* + iay = ; am
termine their points of intersection.
5. Determine the equation to the tangent at L.
Fig. to Art. 126.)
Find the angle between the lines in examples 1 a
7. Determine the equation to the normal at L.
8. Find the point where the normal at L meets the
again, and the length of the intercepted chord.
9. Find the point in a parabola where the tangi
inclined at an angle of 30° to the axis of ar.
i of the perpendicular from the t
the directrix on the tangent at (x, y) is .. , , -^ .
11. Find the po!nt3 of contact of tangents the p
diculars on which from the foot of the directrix are eq
e-fourth of the latus rectum.
EXAMPLES ON THE PAKABOLA. 135
12. A circle has its centre at the vertex A of a parabola
Lose focus is 8 y and the diameter of the circle is 3A8;
3w that the common chord bisects A 8.
13. Trace the curve y = x — x*, and determine whether
B straight line x + y =* 1 is a tangent to it.
14. The tangent at any point of a parabola will meet the
rectrix and latus rectum produced in two points equally
stant from the focus.
15. PM is an ordinate of a point P in a parabola ; a fine
drawn parallel to the axis bisecting PM and cutting the
■rve in Q; MQ cuts the tangent at the vertex A in T;
*w that AT =%PM*
16. If from any point P of a circle PC be drawn to the
litre G, and a chord PQ be drawn parallel to the diameter
CB and bisected in B, shew that the locus of the inter-
ction of CP and AB is a parabola.
17. Find the ordinates of the points where the line
^mx + c meets the parabola; hence determine the ordi-
tte of the middle point of the chord which the parabola
tercepts on this line.
18. A is the origin, B is a point on the axis of y r BQ a
tie parallel to the axis of x ; in A Q, produced if necessary,
is taken such that its ordinate is equal to BQ; shew that
e locus of P is a parabola.
19. From any point Q in the line BQ which is perpen-
sular to the axis CAB of a. parabola whose vertex is A y
Q is drawn parallel to the axis to meet the curve in P;
ew that if 6 A be taken equal to AB, the lines A Q and
P will intersect on the parabola.
20. At the point [x\ y') a normal is drawn ; find the co-
dinates of the point where it meets the curve again, and the
igth of the intercepted chord.
136 EXAMPLES ON THE PAEABOLA.
21. If the normal at any point P meet the curve
in Q, and SP= r, and p he the perpendicular from 8 an
tangent at P, then PQ -^- ,
22. P is any point on a parabola, A the vertex ; tbroi| !£
-4 is drawn a line perpendicular to the tangent at f, «^
through Pis drawn a line parallel to the axis; the lines th
drawn meet in a point Q; shew that the locus of @ii
straight line. Find also the equation to the locus of
the intersection of the perpendicular from A and the ordinal "
at P.
23. PQ is a chord of a parabola, PT the tangent at J
A line parallel to the axis of the parabola cuts the ta-
in T, the arc PQ in E, and the chord PQ in F. Shew
TE : EF :: PF : FQ.
24. In a parabola whose equation is y" = iacc,
tangents are drawn at points whose abscissa? are in tne nti
of 1 : /i ; shew that the equation to the locus of their
section will be
tf ={/** +fL~ i y ax
when the points are on the same side of the axis, ai
y* = -{(J-(j,-*)'aa:
when they are on different sides,
25. Two straight lines are drawn from the vertex
■ parabola at right angles to each other ; the points whef
these lines meet the curve are joined, thus forming a right-
angled triangle ; find the least area of this triangle.
26. Let r and r be the lengths of two radii vectoia
drawn at right angles to each other from the vertex of i
parabola; then
(/r')Ul6a s {r*-r»-' ! ).
27. Find the polar equation to the parabola referred t°
the foot of the directrix as origin and the axis of the curve «»
initial line.
in tharf
their ir*:-l ■
and
e vertex «
EXAMPLES ON THE PARABOLA. 137
If a line be drawn from the foot of the .directrix
; the parabola, the rectangle of the intercepts made by
rve is equal to the rectangle of the parts into which
rallel focal chord is divided by the focus.
Find the polar equation to the parabola when the
: the directrix is the origin and the initial line the
LX.
A system of parallel chords is drawn in a parabola ;
le locus of the point which divides each chord into
its whose product is constant.
S
In a triangle ABC if tan -4 tan— =2, and -42? be
the locus of G will be a parabola whose vertex is A
jus B.
Find the equation to the parabola referred to tan-
it the extremities of the latus rectum as axes.
Find the equation to the parabola referred to the
[ and tangent at L as axes.
P is a point on a parabola ; x\ if are its co-ordinates;
e equation to the circle described on SP as diameter.
Shew that the circle described on SP as diameter
f the tangent at the vertex.
If the line y = m (x — a) meets the parabola in (a?', y ')
", y"), shew that
s" = 2a+g; x'x" = a*; y'+f = %; yY = -*a 9 .
^^A circle is described on a focal chord of a parabola
neterT if^aijbe the tangent of the inclination of this
o the axis of as, the equation to the circle is
\ my * m
I iliaci'V i
138 EXAMPLES ON TIIE PARABOLA.
38. Any circle described on a focal chord as
touches the directrix.
r 39. If the focus of the parabola he the origin, shew I
'' the equation to the tangent at (x, y') is
yy' = 2a {x + x' + 2a).
/40. If the focus of a parabola be the origin, shewl!* t.
the equation to a tangent to the parabola is
y = m(x + a) + ^.
41. Two parabolas have a common focus and axis, aw!
tangent to one intersects a tangent to the other at rifi
angles ; find the locus of the point of intersection.
42. If a chord of the parabola y 1 = Aax be a tan sen' 1
the parabola _y ! = 8a {x — c), shew that the line j;=cfa*
that chord.
43. From any point there cannot be drawn i
three normals to a parabola.
44. In a parabola whose equation is y'=iax, the ordiMM
of three points such that the normals pass through the an*
point are y, y r y a ; prove that y, +,y, + y, = 0. Shew lie
that a circle described through these three points paw
through the vertex of the parabola.
45. If two of the normals which can he drawn to n P*"
bola through a point are at right angles, the locus of th»t
point is a parabola.
46. If two equal parabolas have the same focus andtiW
axes perpendicular to each other, they enclose a Bpace wt#
length PQ = twice the latus rectum, and breadth
latus rectum
47. Find the length of the perpendicular from an ells'"
nal point (A, h) on the corresponding chord of contact
out
EXAMPLES ON THE PABABOLA. 139
48. From an external point (A, h) two tangents are
stwn to a parabola ; shew that the length of the chord of
a tact is
(y + 4rf)*(ff-4at)*
a
49. From an external point (A, k) two tangents are drawn
a parabola; the area ot the triangle formed by the tan-
Qts and chord is - — - — — .
2a
50. Tangents to a parabola TP, Tp are drawn at the
tremities of a focal cnord; P0 7 pg are normals at the
1 1
aae points. Shew that -pp- 9 H - a is invariable ; and that
i normals subtend equal angles at T.
51. Two equal parabolas have the same axis, but their
rtices do not coincide. If through any point on the inner
rve two chords of the outer curve P0p 7 QOq, be drawn
light ancles to one another, then -=7? — 7r + -^-^ — tt is
° ° JrO . up QO . Oq
variable.
52. A circle, described upon a chord of a parabola as
fcmeter just touches the axis ; shew that if be the inclina-
'U of the chord to the axis, 4a the latus rectum of the
irabola, and c the radius of the circle,
/» 2a
tan = — .
c
53. If 0, & be the inclinations to the axis of the para-
la of the two tangents through (A, k), shew that
tan + tan & = t ; tan tan = T .
54. If two tangents be drawn to a parabola so that the
Cn of the angles which they make with the axis, is constant,
140 EXAMPLES ON THE PARABOLA.
the locos of their intersection will be a straight line passing
through the focus.
55. Shew that the two tangents through (A, h) are repi*
sented by the equation
A(y-fc)*-fc(y-fc)(a;-A)+a(a;-A) , = 0;
or (& 2 -4aA) tf-±ax) = {%-2a (x + A)}\
56. Shew that the lines drawn from the vertex to tk
points of contact of the tangents from (A, k) are represent
by the equation
Ay* = 2x (Jcy — 2ax).
£gjMati*n, to tkt Elupbe.
S7^ e* W*
id OS *J>, c**r ( yAn.x c+++1>-< * j~ ?
• , i i it
*f J 1 -^!"^ 1
""- *♦*— JoJ"
.*_ «: /a*-x 4 )
3*5*1
( 141 )
CHAPTER IX.
THE ELLIPSE.
.58. To find the equation to the ellipse.
rhe ellipse is the locus of a point which moves so that its
nee from a fixed point bears a constant ratio to its dis-
3 from a fixed straight line, the ratio being less than
&6>*
liet 8 be the fixed point, FP the fixed straight line.
v 80 perpendicular to YY'; take as the origin, 08
le direction of the axis of x, OY&s that of the axis of y.
jet P be a point on the locus; join SP; draw PM parallel
'Fand PN parallel to OX. Let OS=p, and let e be the
of 8P to PN. Let x, y be the co-ordinates of P.
142
EQUATION TO THE ELLIPSE.
By definition,
SP^e.PN;
.-. PM % + SM* = e»P.tf\
that is,
This is the equation to the ellipse with the assumed
and axes.
159. To find where the ellipse meets the axis of a,
put y = in the equation to the ellipse; thus
(x —p) % = eV ;
.\ a? — j> = ±ex;
__ p
• • x *— "~ ~ •
1 + e
Let OA' = t-t- and OA = -^—; then^and A' mm*
1 + e 1 — e r
on the ellipse.
A and A' are called the vertices of the ellipse, and (J, 4
point midway between -4 and A', is called the centre of th
ellipse.
160. We shall obtain a simpler form of the equitta*
the ellipse by transferring the origin to A' or G.
I. Suppose the origin at A*.
Since OA 9 = rr- , we put x = a?' + -£— and subBifo*
" 1 + e r 1 + e
this value in the equation
y* + (x—j>)* = e?x*;
'♦('-if-M'-if-.)' .
I
Ifcl
»l
k
or
EQUATION TO THE ELLIPSE. 143
/. f = 2pex - (1 - e 9 ) x'*
The distance A A = -^ —*— = --^ , we shall denote
1 — e 1+e 1 — er
* by 2a ; hence the equation becomes
3 f=(l-e s )(2aa5'-a?' 2 ).
We may suppress the accent if we remember that the ori-
. is at the vertex A', and thus write the equation
y* = (l-J)(2ax-a?) (1).
H. Suppose the origin at C, ^
Since A'C—a, we put x = x'+a and substitute this value
(1); thus
3,«= (i-e s ) {2a(x' + a) - (x' + a)*}
= (l-e a )(a 8 -a?' s ).
We may suppress the accent if we remember that the ori-
i is now at the centre C, and thus write the equation
j» - (1 - *) (c* - rf) (2).
In (2) suppose x= 0, then y*= (1 — e 2 )** 2 ; if then we denote ,
t ordinate CB by 6 we have 6* = (1 — e 2 ) a 2 ; thus (1) may
Written
tf-£{aaa-tf) (3),
Cv
3 (2) may be written
?=",{<*-*) • w.
, more symmetrically,
- % +£ = l, or ay + Va? = a % V (5).
FORM OF THE ELLIPSE.
161. Since A'S=eOA' and OA' = — £-, we hive
"-if-.-^-a-*
l + « 6
SC = A'C-A'S=a-a(l-e)-ae,
OO-A'C+OA'-a + ^-^-^i,
162. We may now ascertain the form of the ellij
Take the equation referred to the centre as origin
For every value of x less than a there are two values of
equal in magnitude but of opposite sign. Hence if P I*
FORM OP THE ELLIPSE. 14o
it in the curve on one side of the axis of x there is a point
^n the other side of the axis such that P'M=PM. Hence
curve is symmetrical with respect to the axis of x. Values
b greater than a do not give possible values of y; hence,
being equal to a, the curve docs not extend to the right
a ascribe to x any negative value comprised between
, we obtain for y the same pair of values as when we
Tibed to x the corresponding positive value between
a. Hence the portion of the curve to the left of YY' is
ilar to the portion to the right of YY',
As the equation (1) may be put in the form
see that the axis of y also divides the curve symmetrically
L that the curve does not extend beyond the points B and
where OB and CB' each = b.
The line E'K' is the directrix; 8 is the corresponding
us.
Since the curve is symmetrical with respect to the line
jHT, it follows that if we take CH= GS and CE= GE',
I draw EK perpendicular to GE, the point II and the line
£ will form respectively a second focus and directrix by
bus of which the curve might have been generated.
163. The point Cis called the centre of the ellipse because
fy chord of the ellipse which pauses through C is bisected in
(For suppose (h, k) to be a point on the curve, so that
equation
latisfied by the values x = h, y = k; then (—A, — k) is ahjo .
tint on the curve, because since x = h, y = k, satisfy the
■e equation, it is obvious that x = — h, y = — k, will also
" it. Hence to every point P on the curve there corre-
another point P, in the opposite quadrant, such that
10
z
146 FORM OF THE ELLIPSE.
PGP t is a straight line and P t C*=PC. Hence every cl
passing through G is bisected in C.
164. We have drawn the curve concave towards the
of x; the following proposition will justify the figure.
The ordinate of any point of the curve which lies bet
a vertex and a fixed point of the curve is greater than
corresponding ordinate of the straight line joining that
and the fixed point.
Let A be the vertex, and take it for the origin; letPl
the fixed point ; x\ rj its co-ordinates. Then the eqi
to the ellipse is (Art. 160)
a '
The equation to A'P is y = *L Xf or y = — . / [-r-lk
since {x\ y') is on the ellipse.
Let x denote any abscissa less than x, then since tk
ordinate of the curve is -»J(2ax — of) or -a/(~~- 1 P
and that of the straight line is - i/[—t — 1 ) x, it is obviw
that the ordinate of the curve is greater than that of 4*
line.
165. A A' and BB are called axes of the ellipse. Tk
axis A A' which contains the two foci is called the major axis
and sometimes the transverse axis; BE is called the mitf
axis and sometimes the conjugate axis.
The ratio which the distance of any point in the ellif*
from the focus bears to the distance of the same point fi*
the corresponding directrix is called the excentricitfy of tk
ellipse. We have denoted it by the symbol e. ;$k
To find the latus rectum (see Art. 128) we put N \=CR
that is, = ae, in equation (1) of Art. 162; thus
,_ &y(i-e') y.
FOCAL DISTANCES OP ANT POINT. 147
.*. LH= — , and the latuS rectum = — .
a 9 a
nee J'aa'-aVj .\ 5* + aV = a f ; that is,
CB*+CH* = a*;
.: BH=a\
xly , BS = a.
S6. * To express the focal distances of any point oj ? the ^
? in terms of the abscissa of the point.
et She one focus, E'K' the corresponding directrix ; H
fcher focus, EK the corresponding directrix. Let Pbe a
on the ellipse; x,y its co-ordinates, the centre being
rigin. Join SP, HP, and draw N*PN parallel to the
* axis, and PM perpendicular to it.
hen SP=ePN* = e{E'C+CM) = efi+x\ = a + ex.
Iso, EP=ePN=e(CE-CM) = e(Z-cc\ = a-ex.
148 EXCENTRIC ANGLE.
Hence SP+ SP= 2a; that is, the sum of the focal
tancea of any point 011 the ellipse is equal to the major as
1 67. The equation y 1 = -, (a 1 — 3?) may be written
- a (a~x)(a + x).
Hence (see Fig. to Art. 162)
PM* _BC
A'M.MA AC
168. Let a circle be described on the major axis of
ellipse as a diameter; its equation referred to the centre
origin will be
y'=a*-x\
Hence if any ordinate MP of the ellipse be produced to n
the circle in P 1 we have
PM _b
' P'M a ■
Join P with C the centre of the ellipse ; let P
and let x, y be the co-ordinatea of P; then
x= CP cos <f> = a cos <j>,
PCU-
These values of * and y are sometimes useful in the i
tion of problems.
CONNEXION OF TfiB ELLIPSE AND PARABOLA. 149
The angle P'CM is called the excenirto angle of the
oint P.
169. From Art. 160 we see that the equation to the
Uipse when the vertex is the origin is
y 2 3= 2pea? — (1 — 4?) a*.
If we suppose e = 1, this becomes
y* = 2px f
"hich is the equation to a parabola whose latus rectum is 2p.
Also in the ellipse
«£_
6
•(1
vert
TANGENT TO AN ELLIPSE.
If we now make e = l, we have o and b infinite, i
(1 — e) =^ . Thus if we suppose the distance between 1
vertex and nearer focus of an ellipse to remain consti
while the excentricity approaches continually nearer to nni
the major and minor axc3 of the ellipse increase indefinite
and the ellipse about the vertex approximates to the fa
of a parabola.
Thus if any property is established for an ellipse we m
seek for a corresponding property in the parabola by referrii
the ellipae to the vertex as origin and examining what tl
result becomes when e is made to approach continually
inity, while the distance between the vertex and the new
focus remains constant.
Tangent and Normal to an Ellipse.
the tangent at any po&*
%- 170. To find the equation
an ellipse. (See Def. Art. 90.)
Let x', y be the co-ordinates of the point,
x",y" the co-ordinates of an adjacent poii
r
The equation to the secant through these points ti
y-y'-^i^-'l) CI;
«ince (x, y') and (;
y") are points on the ellipse,
aY+b*x 1 = a t b;
a^y" 1 + £V = aVf ;
.-. a* fo^-yj + B 1 (jb"»-<O-0;
a' y +y
TANGENT TO AN ELLIPSE. I5J
Hence (1) may be written
v — y= — s. —77-: — Ax — x\+
J * a 9 y " + y' v /
Now in the limit x" = x\ and jf ij "*tf ; hence the equation
[> the tangent at the point {x\y) is
l*x
j^=-—( x -x') :.(»)•
XUb equation may. be simplified ; multiply by a'y'> thus
a*yy' + Vxx' = a 9 / 8 + b V 2 = aV.
171. The equation to the tangent can be conveniently
(.pressed in terms of the tangent of the angle which tli
ae makes with the major axis of the ellipse.
For the equation to the tangent at (x f y') is
a'yy' + W-a 2 } 1 ,
JV >
r y= rix + — .
ay y
Let j- t = 01 ; thus the equation becomes
y=mx + -p;
e hare then to express —, in terms of m.
Now JV = — (fy'm,
ad W+W-tW;
, „ ditty 1 * ...
.'. if'*(c?m*+b*) = b\
,\ ? -,/(«*■* + «.
152 NOEMAL TO AN ELLIPSE.
Hence the equation to the tangent may be writfc
y = mx + *J (aVw' + S 1 ).
Conversely every line whose equation is of this form
a tangent to the ellipse.
It may be shewn as in Arts. 93, 94, that the tangent*
any point of an ellipse meets it in only one point, and M
a line which meets an ellipse in only one point is the tangti
at that point.
172. The tangents at the extremities of either axis
parallel to the other axis.
For the co-ordinates of A are a, 0. (See Fig. to Ait. ifii,
Hence, putting x' = a, y' = 0, the equation
which is the equation to a line through A parallel to CI
Similarly the tangent at A' is parallel to C'Y, and the tan-
gents at B and B' are parallel to CX.
173. To find the equation to the normal at any point of
an ellipse. (See Def. Art. 97.)
Let x, y be the co-ordinates of the point; the equanV
to the tangent at that point is
■y
The equation to a line through [x, y) perpendicular u>
(i)i»
y-/-0§(»-«O
This is the equation to the normal at (a:', y').
NORMAL TO AN ELLIPSE. 153
m 174. The equation to the normal majr also be expressed
terms of the tangent of the angle which the line makes
th the major axis of the ellipse.
1 The equation to the normal at (x\ y) is
oV fa* A ,
2|Let jr-, = m ; thus the equation becomes -
Vx
a 1
/»# ^— imnm» —
y = mx jz— y (1);
**' .a vt
a if
3 have then to express — 75 — y* in terms of m.
.«-.'
THoir, JV-^t,
jfc .-. y* (bW + a*) =bW.
'■' Hence (1) becomes
(a* -&*)»»
«» y — ww? — //ix » , *\ •
. 175. We shall now deduce some properties of the ellipse
im the preceding articles.
Let x\ y* be the co-ordinates of P; let PTbe the tangent
I P, and P# the normal at P; PJf, PN perpendiculars on
ieaxes.
. The equation to the tangent at P is
a*yy' + b*xx' = a'b*.
a
Let y = 0, then x=—,, hence
x
CM'
154
PROPERTIES OF THE ELLIPSE.
.•. CM.CT=CA*.
Similarly, if the tangent at P meet CY in T\
CN. CT = OB.
176. The equation to the normal at P is
At the point O where the normal cuts the maj<
y = 0, hence from the above equation
a?— x = —
a
* t
•\ x = a?' ( 1 — 5 j = eV.
Thus CG^e'CM.
At the point (?' where the normal cuts the min
x = 0, hence from the above equation
Thus
CO' = ^PM.
PROPERTIES OF THE ELLIPSE. 155
r. The lengths of PG and PG' may be conveniently
aed in terms of the focal distances of P.
PG* = PM'+GM*
-y»+ar»(i-/)»
* a
a x 'a
-|k-(.-5-i
i &P= r', J5GP=» r ; then
,, Vrr
a*
lilarly, it may be shewn that
p&"=^-
I. The normal at any point bisects the angle between ^
il distances of that point.
; x\ y* be the co-ordinates of P; the co-ordinates of 8
e, ; hence the equation to SP is (Art. 35)
y = -rr — (x + ae) (1).
; x + ae v ' . v ' .
3 equation to the normal at P is
156 PERPENDICULAR ON THE TANGENT.
Hence the tangent of the angle OPS
«y y
_ Wx x' + as _ {<£ ! — V) x'y' + a*ey'
- , aY ay + b*x»+b*x'ae
+ iV(a?' + o«)
— <*y a?y 4- oty ««y
~ a*b* + b*x'ae ""F - ,
The equation to IZP is
hence it may be shewn that the tangent of the angle (H
also = ~- ; |
.-. spg=hpg: I
Hence SPT = HPT; that is, the tangent at any poU
equally inclined to the focal distances of that point
179. The preceding proposition may also be estabtt
thus:
CG = eV, (Art. 176) ;
,\ SG = ae + £x',
and HG = ae — £x\
Also SP=a + ex\ HP=a — ez; hence
8G SP.
HG~HP>
therefore by Euclid, vi. 3, PG bisects the angle 8PB.
180. To find the locus of the intersection of the tanjpt
any point with the perpendicular on it from the focus.
Let y = wia? + V(&* + Wtf) fl!
be the equation to a tangent to the ellipse (Art 171); ^
the equation to the perpendicular on it from the focus H
(see Fig. to Art. 175)
y = (x-ae) ft
PERPENDICULAR ON THE TANGENT. 157
f we suppose x and y to have respectively the same
m in (1) and (2), and eliminate m between the two equa-
i, we shall obtain the required locus.
Trom (1) y — mx = *J(b* + mV) ;
(2) my + x = ae;
re and add, then
(y* + af) (1 + m*) = b* + mV + a V
= a 1 (l + m*);
•v y*+a? = a %
*
be equation to the required locus, which is therefore a
e described on the major axis of the ellipse as dia-
We have supposed the perpendicular drawn from H\ we
I arrive at the same result if it be drawn from 8; hence
2?, 82f be these perpendiculars, CZami CZ each = a.
181. To find the length of the perpendicular from the focus ^L
he tangent at any point.
The equation to the tangent at the point (x, y) is
iV , b*
Ihe co-ordinates of the focus if are ae, 0. But if p denote
length of the perpendicular from a point (a\, yj-on die
yssmx + c, by Art. 47
-^ 1 + m*
In the present case
m = — 5—, , c = -7 ;
ay y
TWO TANGENTS FROM AN EXTERNAL POINT.
Since t
b*x" a'y'* + 6V*
a'b^a-exy «V(o-
- f j-y
a' (oV - JV") + 5 V a" {a* —
,v,
a + ex r v ■
a — r we have » = .
Similarly if ^' be the perpendicular from 8 on the tang ,"
at [x', y) we shall find
* -— ;
182. J^j-ont aMy external point two tangents can fa i
to an ellipse.
Let the equation to the ellipse be
ay + Fx'=a*b t (Ir
and let h, k be the co-ordinates of an external point. 3
* pose x, y the co-ordinates of a point on the ellipse, i
that the tangent at this point passes through (k, k). B
equation to the tangent at (x\ y) is
a*yy' + b t xx' = a t b*
Since this tangent passes through (h, k)
a*ky' + Vhx' = a'o* (!)■
Also since (x, y) is on the ellipse
ay+b'x'^aV [l\
Equations (3) and (4) determine the values of x' and;'.
CHORD OF CONTACT. 159
Substitute from (3) in (4), thus
x n (aW + VK) - 2c?Vhx' + a\V - V) = 0.
* •
Che roots of this quadratic will be found to be both pos-
since (A, h) is an external point and therefore a*I£ + Vtf
tcr than a*b*.
[The line which passes through the points where these
ents meet the ellipse is called the chord of contact
.83. Tangents are drawn to an ellipse from a given exter-, r
taint; to find the equation to the chord of contact.
liet h, h be the co-ordinates of the external point; x t9 y x
20-ordinates of the point where one of the tangents from
s) meets the ellipse; x t , y % the co-ordinates of the point
re the other tangent from (n, k) meets the ellipse.
Che equation to the tangent at (x v y t ) is
a t yy l + b 9 xx l = a 9 b' i (1);
* this tangent passes through (h, k) we have
a%, + b*hx x = aV (2).
Similarly, since the tangent at (x t , y 2 ) passes through
a*ky t +b*hx % =a*b*... (3). *
Eence it follows that the equation to the chord of contact is
a*ky + b°hx = a 2 b*. (4).
•
?or (4) is obviously the equation to some straight line
this line passes through (x x , y x ) for (4) is satisfied by the
ea x = x t , y~y x as we see from (2); similarly from (3)
onclude that this line passes through (x t , y t ). Hence (4)
•e required equation.
Fhus we may proceed as follows in order to draw tangents
>n ellipse from a given external point— draw the line
) CHORD OF CONTACT.
which ia represented by (4) ; join the points where it
the ellipse with the given external point, and the line
obtained are the required tangents.
184. Through any fixed point chords are dram
ellipse, and tangents to the ellipse are drawn at tlie extr
of each chord/ the locus of the intersection of the tanget
straight line.
Let h, k be the co-ordinates of the point through
the chorda are drawn ; let tangents to the ellipse he ilr
the extremities of one of these chords, and let (x t , y t )
point in which they meet. The equation to the a
chord of contact is, by Art. 183,
a'yy l + b'xx 1 = a'b'
But this chord passes through (h, k) ; therefc
a*ky l + b t hx i = a'b*.
Hence the point (x l , y,) lies on the line
a'ky + b'hx = a'b' ;
;
that is, the locu3 of the intersection of the tangen'
straight line.
We will now prove the converse of this proposition,
185. If from any point in a straight line a pair
gents be drawn to an ellipse the chords of contact will t
through afixedpoint.
be the equation to the straight line ; let (x' t y) be a p
this line from which tangents are drawn to the ellipse
the equation to the corresponding chord of contact is
a'yy' -f b'xx' = a'b*..
Since (*',y)i. on (1)
AzZ + By'+C-O;
INTERPBETATI0N8 OF AN EQUATION. 161
ore (2) may be written
6*a%e d — ay = ab,
^_^) x '_^_ o .J. = (3).
ow, whatever be the value of x\ this line passes through
oint whose co-ordinates are found by the simultaneous
ions
s, the point for which
BV _ Atf
y -Q> *-- 6 r •
$6. The student should observe the different interpre-
ts that can be assigned to the equation
cfky + b'hx = a*b*.
he statements in Art. 103 with respect to the circle may
I applied to the ellipse.
EXAMPLES.
What is the excentricity of the ellipse 2jb* + 3y" = c* ?
Find* the equation to the tangent at the end of the
rectum L. (See Fig. to Art. 162.) Also find the lengths
i intercepts of this tangent on the axes.
Write down the equation to the normal at L.
, If the normal at L passes through the extremity of
linor axis B\ what is the excentricity of the ellipse?
. Find the equations to A'B and CL. (See Fig. to Art.
What is the excentricity of the ellipse if these lines fure
lei?
C. S. W
I
162 EXAMPLES ON THE ELLIPSE.
6. Find the equation to if H, and determine the al
of the point where this line cuts the ellipse again.
7. Find the equation to Ah, and determine
between this line and the tangent at L.
8. If from the point P whose abscissa is x', a, li
drawn through H, determine the abscissa of the point vhs
it meets the ellipse again.
' 9. Find a point in the ellipse such that the tangent !li
is equally inclined to the axes.
'10. Find a point in the ellipse such that the intei
made by the tangent on the co-ordinate axes are proportion!
to the corresponding axes of the ellipse.
11. Pis a point on an ellipse, y its ordinate; she?
26'
12. Pis a point on an ellipse, y its ordinate; shewth
the tangent of the angle between the focal distance audi
tangent at P is — .
aey
13. If <£ denote the angle mentioned in the preceding 1
question,
PC = *J {a' -b' cot* <f>
14. From P a point in an ellipse lines are drawnto A, A,
the extremities of the major axis, and from A, A! lines an
drawn perpendicular to AP, A'P; shew that the locus of their
intersection will be another ellipse, and find its axes.
15. If any ordinate MP be produced to meet the tangent
at L in Q, prove that QM= PH. (See Fig. to Art. 162.)
16. If a aeries of ellipses be described having the same
major axes the tangents at the ends of their latera recta will
pass through one or other of two fixed points.
tan APA' - -
EXAMPLES ON THE ELLIPSE. 163
. If the focus of an ellipse be the common focus of two
olas whose vertices are at the ends of the axis major,
parabolas will intersect at right angles, at points whose
ce from each other is equal to twice the minor axis.
. Shew that the length of the longer normal drawn
a point in the minor axis of an ellipse at a distance c
the centre and intercepted between that point and the
is
. If any parallel straight lines be drawn from the focus
i the extremity A of the axis major of an ellipse, and
ind N be the points where they meet the axis minor, or
us minor produced, then the circle whose centre is M
idius NA will either touch the ellipse, or fall entirely
e of it.
. A and A* are the extremities of the major axis of an
j, T is the point where the tangent at the point P of
urve meets A A' produced ; through T a line is drawn
idicular to AA' and meeting AP and A'P produced in
. R respectively; shew that QT=RT.
. If <f>, <f>' be the excentric angles of two points, th$
on to the chord joining the points is
x d>-\-d> y . d> + <b 4> — <fr
— cos T -^ J - + r Bin r ^ = cos ^ .
a 2 b 2 2
. Express the equation to the tangent at any point in
of the excentric angle of that point.
. Shew that the equation to the normal at the point
excentric angle is <f> is
ax sec <f> — by cosec <f> = a* — b*.
. The locus of the middle point of PO (see Art. 176) is
pse of which the excentricity e' is connected with that
given ellipse by the equation
. l-e'^l+^lW 2 ).
11— a
164 EXAMPLES ON THE ELLIPSE.
25. Determine the point of intersection of the tangent j
L with the line HB ; what is the value of the escei
the ellipse when these lines are parallel?
26. A tangent at any point P of au ellipse meets th(
directrix EK in T and E'K' in T\ shew that TE
the cotangent of PBS and T'E' varies as the cotangent «
P8H. (See Fig. to Art. 162.)
* 27. If the straight \raty = mx + c intersect i!
tfy* 4- 5V = a 2 b', shew that the length of the chord will be
2aM(l + M°)( W V + fi , - C ')}
mV + J'
Hence find tlie relation between the constants that this lb;
may be a tangent to the ellipse.
28. Find the equation to the circle described c
diameter, supposing x, y the co-ordinates of P.
29. Shew that any circle described on HP as diameter,
touches the circle described on the major axis as diameter.
30. From a point (h, k) two tangents are drawn to ta
ellipse; find the sum of the perpendiculars from the foci on
the chord of contact.
31. Any ordinate PM of an ellipse is produced to
the circle on the axis major in Q and normals to the ellipM
and circle at P and Q respectively meet in R ; find the locus
of J?.
32. Two ellipses have a common centre and their as»
coincide in direction; also the sum of the squares of
is the same in the two ellipses ; find the equation to a common
tangent.
- 33. If 8, f be the inclinations to the major axis of ihe
ellipse of the two tangents that can be drawn from the
(A, %), shew that
tanfl + tanfl' ^* tan tan ^ - -
cm the poini
*]
EXAMPLES ON THE ELLIPSE. 165
"* 34. Find the locus of a point such that the two tangents
=*from it to an ellipse are at right angles.
35. Shew that the two tangents which can be drawn to
s-an ellipse through the point (A, k) are represented by
or by
: J (a 8 £ 8 + i 8 A 8 -a 8 i 8 ) ((ty + VaP-<W) = (a% + Vhx - a*b % )\
Fi
36. Tangents are drawn to an ellipse from the point (A, k) ;
shew that the lines drawn from the origin to the points of
contact are represented by
a 8+ J 8 ~U* W'
* 37. Pairs of radii vectores are drawn at right angles to
each other from the centre of an ellipse ; shew that the tan-
^ gents at their extremities intersect in the ellipse
•: 38. From an external point T whose co-ordinates are A
aijd k a line is drawn to the centre C meeting the ellipse in
JS, shew that
GT % a*k* + b*h*
212
CR* a a 6
39. From an external point (A, k). tangents are drawn; if
x l9 x % be the abscissae of the points of contact, shew that
2ka*b* _ a'(b*-J<! 1 )
40. From an external point (A, k) tangents are drawn
meeting the ellipse in P and Q ; find the value of HP. HQ,
H being a focus.
41. From an external point T the lines TP, TQ are
drawn to touch the ellipse in P and Q. CT cuts the ellipse
166 EXAMPLES ON THE ELLIPSE.
in R, and RN is drawn parallel to HT to meet the a
in H; shew that HP. H Q = RN\
42. Two ellipses of equal excentricity and whose DM
axes are parallel can only have two points in common. Pro!
this, and shew that if three such ellipses intersect, two and
two in the points Pand P', Q and Q\ R and H', respectively,
the lines PP, QQ', MR 1 , meet in a point.
43. Two concentric ellipses which have their axes in th
Bame direction intersect, and four common tangents are drawl
so aa to form a rhombus, and the points of intersection of tin
ellipses are joined so as to form a rectangle ; prove that tin
product of the areas of the rhombus and rectangle is equal V>
half the continued product of the four axes,
44. If the ordinate at any point P of an ellipse be
daced to meet the circle described on the major axis **
diameter in Q, prove that the perpendicular from the focus >
on the tangent at Q is equal to SP.
45. Find the equation to the ellipse referred to axe
passing through the extremities of the minor axis, and meet-
ing in one extremity of the major axis.
46. If from points of the curve — , -f -j = (a 1 — £*)', tangent)
be drawn to the ellipse -„ 4- "4 = 1, the chords of contact nil
r a" o
be normal to the ellipse.
47. Prove the proposition in Art. 180 in a manner sirniLtf
to that used in Art. 138. Also prove the proposition in Art
138 in a manner similar to that used in Art. 180.
48. Find the equation to the ellipse the origin b"
point (h, k) on the ellipse and the axes parallel to the axes of
the ellipse.
49. From a point P on an ellipse two chords PQ t PQ are
drawn meeting the ellipse in Q, Q'; if h, k be the co-ordi-
nates of P referred to the centre, and mx + ny = 1 the eqt
EXAMPLES ON THE ELLIPSE. 167
o QQ' referred to P as origin, shew that the lines PQ, PQ'
re represented by
rith P as origin.
50. Let P be any point on an ellipse ; draw PP f parallel
the major axis and cutting the curve in P'; through P draw
pro chords PQ, PQ', making equal angles with the major
ixis ; join QQ' ; QQ shall be parallel to the tangent at P'.
51. From the equation y ■* mx + V(wV + 6 2 ) deduce the
quation to the tangent to the parabola.
52. In the figure of Art. 175 suppose GP produced to a
point Q such that GQ = n. GP, and lind the locus of Q.
'■ H3. If PN be any ordinate of a circle, and from the ex-
remity A of the corresponding diameter AB, A Q be drawn
neeting PN in Q, so that A Q = PJV", find the locus of Q and
he position of its focus.
54. Express the tangent of the angle between GP and
lie normal at P in terms of the co-ordinates of P.
55. Find the greatest value of the tangent of the angle
letween GP and the normal at P.
56. The major axis of an ellipse is equal to twice the
ainor axis ; a line of length equal to half the major axis is
►laced with one end on the curve and the other on the minor
atis ; shew that the middle point of the line is on the major
57. A circle is inscribed in the triangle formed by two
Deal distances and the major axis of an ellipse ; find the locus
f the centre.
58. If BZ\ HZ be perpendiculars on the tangent at the
K>int P of an ellipse, SZ and HZ' will intersect on the normal
XP.
( 168 )
CHAPTEE X.
THE ELLIPSE CONTINUED.
Diameters.
187. To find the length of a line drawn from anypoii
a given direction to meet an ellipse.
Let x\ y be the co-ordinates of the point from whicl
line is drawn; x, y the co-ordinates of the point to v
the line is drawn ; the inclination of the line to the ax
x ; r the length of the line ; then (Art. 27)
x = x f +r cos 0, y = y+rsm0 (:
If (x, y) be on the ellipse these values may be substi
in the equation
ay + b*x* = a*b*; thus
a* (y' + r &in0)*+b* (x' + r cos0y= a*b* ;
.'. r 8 (a 2 sin 9 + b* cos 2 0) + 2r (a*y' sin + b*x' cos 6)
+ a*y'*+ &V 2 -a 2 J 2 = (
From this quadratic two values of r can be found whic
the lengths of the two lines that can be drawn from (i
in the given direction to the ellipse.
188. To find the diameter of a given system of pa
cJiords in an ellipse. (See definition in Art. 148.)
Let be the inclination of the chords to the major a
the ellipse ; let x\ y be the co-ordinates of the middle
of any one of the chords ; the equation which determine
DIAMETERS OF THE ELLIPSE. 169
:1s of the lines drawn from (x, y') to the curve is
. 187)
sin 2 + b* cos 2 0) + 2r (a*y' sin + b*x' cos 3)
+ ay a + JV 2 -a 8 J 2 =0 (1).
3 ( x > y) I s the middle point of the chord, the values of r
shed by this quadratic must be equal in magnitude and
9tte in sign ; hence the coefficient of r must vanish ; thus
b*
sin0 + JVcos0 = O, or y ' = 5 cot0.a;' '(2).
Considering x' and y' as variable, this is the equation to a
ght line passing through the origin, that is, through the
te of the ellipse.
Hence every diameter passes through the centre.
Eso every straight line passing through the centre is a
aeter, that is, bisects some system of parallel chords ; for
giving to a suitable value the equation (2) may be made
bpresent any line passing through the centre.
If ff m be the inclination to the axis of x of the diameter
ch bisects all the* chords inclined at an angle we have
*(2) ■ ,;':-^V--
tan 0' = — 5 cot ;
<> . - «
a
.\ tan tan 0'=---, ...-........(3).
189. Tf one diameter bisect all chords parallel to a second
neter, the second diameter will bisect all chords parallel to
first.
"Let X and 8 be the respective inclinations of the two
meters to the major axis of the ellipse. Since the first
sets all the chords parallel to the second, we have
tan Q tan 0. = — , .
8 l a
170 CONJUGATE DIAMETERS OF THE ELLIPSE.
And this is also the only condition that must hold in
that the second may bisect the chords parallel to the first
190. The tangent at either extremity of any diameter
parallel to the chords which that diameter bisects.
Let A, A; be the co-ordinates of either extremity of
diameter ; the inclination to the major axis of the elH
of the chords which the diameter bisects. Then the W
x = h, y = k must satisfy the equation j
a*y sin + Vxcos = 0;
/• tan o = — ,7 .
a*k
But, by Art. 170, the equation to the tangent at (A,i)i
Hence the tangent is parallel to the bisected chords.
191. Def. Two diameters are called conjugate whenei
bisects the chords parallel to the other.
From Art. 190 it follows that each of the conjugate &
meters is parallel to the tangent at either extremity oft
other.
192. Given the co-ordinates of one extremity of a diixM
to find those of either extremity of the conjugate diameter.
JjetACA',BCB' be the axes of an ellipse; PCP\M
a pair of conjugate diameters.
Let x\ y' be the given co-ordinates of P; then the eqt
tion to CP is
y=^ x W
Since the conjugate diameter DD' is parallel to the tangent
P, the equation to DD' is
Vx ,.i
y=-^ x *
CONJUGATE DIAMETERS OP THE ELLIPSE.
171
mst combine (2) with the equation to the ellipse to
Hordinates of D and D\ Substitute from (2) in
b* at*
ay
' a*b* ~ b* '
•'• & = i ~j~ )
bx
.*. from (2) y = + — .
figure the abscissa of D is negative and that of D'
lence the upper sign applies to D' and the lower
roperties of the ellipse connected with conjugate
are numerous and important; we shall now give
iem.
172 CONJUGATE DIAMETERS OF THE ELLIPSE.
193. The sum of the squares of two conjugate sem
meters is constant.
Let x\ y' be the co-ordinates of P; then by the pi
article
CP* + CD* = x* + y* + ^ + ?^ r
" b a
&» + o 1
= a* + b\
m
Thus the sum of the squares of two conjugate semi-ch
ters is equal to the sum oi the squares of the semi-axes.
Moreover
CI? = a* + b*-x*-y'* = a* + b*-x*-- % (a*-x' t )
= a*-U- h ^x ,2 = a*-e*x*=SP.HPhyk&\
194. The area of the parallelogram which touches them\
at the ends of conjugate diameters is constant.
Let PGP\ DGD' be the conjugate diameters (seeF(
Art. 192). The area of thp parallelogram described sol
touch the ellipse at P, D, P', D\ is AGP. CD sin PCfl
±p.CD, where p denotes the perpendicular, from Con
tangent at P. Let x', y be the co-ordinates of P; thes
equation to the tangent at P is
Vx' , V
y = — 5-1 1 x+—,.
* ay y
V
Hence (Art. 47) p= / , ^ + ^ = V(a y ^V) '
;. Ap.CD = 4a5.
Thus the area of any parallelogram which touches
ellipse at the ends of conjugate diameters is equal to thei
K.PENDICULAR FROM THE CENTRE ON THE TANGENT. 173
e rectangle which touches the ellipse at the ends of the
. Let «', V denote the lengths of two conjugate semi-
^ters; a the angle between them; by the preceding
a'V sin a = ah ;
, _^W 4a 2 y _ 4a 2 5 8
^ a " a"&'* ~" (a' 2 + 6' 2 ) 2 - (a' 2 - b' 2 ) 2 (a* + J 2 ) 2 - (a' 2 - 6' 2 ) 2 '
Cence sin 2 a has its least value when a = b', and then
2ab
B6. From Art. 194 we have
,_ a 2 J 2 _ o»y
' ~~Clr~a* + b*-CP* ( ''
I^his gives a relation between ^> the perpendicular from the
I* on the tangent at any point P and the distance CP of
■point from the centre.
We may also express p in terms of the angle its direction
PS with the axis major ; for let ty denote the angle, then
Equation to the tangent at (a/, y) may be written
a*yy' + b 2 xx' = a 2 b 2 ,
ilso in the form (Art. 20)
x cos ty + y sin yfr =p.
V _ °* P _ a * .
sin t|t ~" y 9 cos ^r x' '
^e
, a J 2 sin -\!r
••• «y — ^s
, , ^_ a 2 J cos yfr ^
OX ^5 .
818
and .'. a 9 b 2 = — ^-{b 2 sin 2 ^ft + a 2 cos 2 yjr);
/. j? = J* sin 2 ->|r + o 2 cos 2 yfr
==a 2 (l-e 2 sin 2 ^r). .
176 TANGENTS AT THE EXTREMITIES OF A CHORD.
This shews that the equal conjugate diameters are
to the lines BA and BA\ ■'
200. The equation to the tangent to the ellipse will hi
the same form whether the axes be rectangular or the obtf
system farmed by a pair of conjugate diameters ; for the!
vestigation of Art. 170 will apply without any change to(
equation a % y % + b^a? = a"b* which represents an ellipse
ferred to such an oblique system.
201. Tangents at the extremities of any chord of an
meet in the diameter which bisects that chord.
Refer the ellipse to the diameter bisecting the chord as
axis of x 9 and the diameter parallel to the chord as
of y ; let the equation to the ellipse be
Let x\ y* be the co-ordinates of one extremity of the
then the equation to the tangent at this point is
a*yy' + b'*xx' ~ rt^.." (\\\
The co-ordinates of the other extremity of the chord
x\ —y\ and the equation to the tangent there is
-a*yy' + b , *xx' = a*b'* ft
The lines represented by (1) and (2) meet at the
for which
_/t
y = o,
this proves the proposition.
a,
X — T t
X
Supplemental chords.
202. Def. Two straight lines drawn from a point
ellipse to the extremities of any diameter are called
mental chords. They are called principal supplemental
if that diameter be the major axis.
SUPPLEMENTAL CHORDS.
177
203. If a chord and diameter of an ellipse are parallel, the
fOfplemental chord is parallel to the conjugate diameter*
Let PP be a diameter of the ellipse ; QP, QP two sup-
©mental chords. Let a>', y' be the co-ordinates of P, and
csrefore —x\ — y' the co-ordinates of P\
Let the equation to PQ be (Art. 32)
y-y' = m (x-x') (1),
t*d the equation to P Q
y + y'=m' (x + x') (2).
The co-ordinates of the point Q satisfy (1) and (2) ; if then
^ suppose x 9 y to denote those co-ordinates, we have from
} ana (2) by multiplication
yi-y'^mm'iaf-x'*) (3).
But since (x, y) and (x, y') are points on the ellipse
a*tf + b V = a 2 b\
••• a*(y 2 -y'*)+b*(x*-x'*)=0;
/.jf-y*«-£(rf-rf") (4);
T. C. S. Vi
178 POLAE EQUATION TO THE ELLIPSE.
From (3) and (4) we have
mm = — 5
a
But we have shewn in Art. 188 that if (5) be satisfiei
two lines represented by y = mx and y = m'x are conj
diameters ; this proves the theorem.
Polar Equation.
204. To find the polar equation to the ellipse, the
being the pole.
Let SP=r, A'SP=0, (see Fig. to Art. 158) ;
then SP = ePN, by definition j
that is, SP=e{OS+SM);
or r = a (1— e 8 ) +ercos(7r — 0), (Art)
/. r (1 + e cos 0) = a (1 — e*),
, a(l-e*)
and r: =rr ~h*
1 4- e cos
If we denote the angle ASP by 0, then we have as b
JSP=e{OS+SM);
thus r = a (1 — e 2 ) + er cos 0,
A a ( l - ')
and r = — * ^ .
1 — 6 cos u
205. We shall make use of the preceding article in fin
the polar equation to a chord, from which we shall deduo
polar equation to the tangent.
Let P and P' be two points on the ellipse ; suppose
A'SP=a-j3, A'SP = a + fr
so that P8P = 2£ ; and let I be the semi-latus rectum oi
POLAE EQUATION TO A CHORD.
179
pse, so that I — a (1 — e*) ; it is required to find the polar
.ation to the line PP.
fume for the equation (see Art. 29)
.4rcos0 + .Brsin0+(7 = O (1).
Since the line passes through P, equation (1) must be
Lafied by the co-ordinates of P; now A'SP=a — j3, and
I
arefore #P= ; -. ^r: thus from (1)
1 + e cos (a — p) v '
1 cos (a-/3) + .Bsin (a- 0)}
+ ^{l + ecos (a -/3)}=0 (2).
Similarly, since the line passes through P\
£ cos (a + £) + .B sin (a + £)}
+ C r {l+ecos(a + j8)}=0 (3).
>m (2) and (3), by subtraction,
Z(i4sinasin/?— -Bcosasin/S) + Cfesin asin/8 = 0;
.'. ?(.4sina-l?cosa) + Cfesina = (4).
Mn (2) and (3), by addition,
(-4cosacos£ + jBsinacosj8) + (7(1 +ecosacosj8) = 0;
.•. I {A cos a + J5sin a) + C (sec £ + e cos a) = (5).
YL—*L
180 POLAR EQUATION TO A TANGENT.
From (4) and (5) we find
*
IA + (sec £ cos a + e) = 0,
IB+ C7secj8sina =0.
Substitute the values of A and 5 in (1) and divide by C
and we have
r {(sec £ cos a + e) cos + sec £ sin a sin 0} — Z = 0;
Z
• fm mm— ___^ __________ ________
ecos0 + sec£cos(a — 0) "
If ## bisect the angle P&P', we have
i°## = A and A'SQ = a.
Now suppose {J to diminish indefinitely ; then the ckflj
PP becomes the tangent at Q, and we obtain its polar equate
by putting {J = in the preceding result ; thus we have
I
r =
e cos + cos (a — 0) *
The investigations of this article will apply to the p«*
bola by supposing 6 = 1.
206. The polar equation to the ellipse referred to
centre is sometimes useful; it may be deduced from
equation a*y a + JV = a 2 5 2 , by putting rcos0, rsinfl, fa 1
and y respectively ; we thus obtain
r a (a a sin 2 + J 2 cos 9 0)=a 2 J 2 .
We add a few miscellaneous propositions on the ellipse.
207. If tangents be drawn at the extremities of anjj
chord of an ellipse, (1) the tangents will intersect in theaf
sponding directrix, (2) the line drawn from the point of
section of the tangents to the focus will be jperpmdictdar^ 1
focal chord.
TANGENTS AT THE EXTREMITIES OP A FOCAL CHORD. 181
(1) If two tangents to an ellipse meet in the point (A, h)
equation to the chord of contact is, by Art. 183,
a*ky + b*hx = a*b\
Suppose the chord passes through the focus whose co-ordi-
ss are x = — ae, y = ; then
— Vhae = a 8 6 2 ,
.\ h = — ;
e
fc is, the point of intersection of the tangents is on the
^ctrix corresponding to this focus.
(2) The equation to the line through (A, h) and the focus is
y — i (x + ae).
* h + ae^ l
If h = — , this becomes
e
Jca* . N
3 the line is therefore perpendicular to the focal chord of
ich the equation is
y to , b*
V- a 2 k + k*
208. If through any point within or without an ellipse, two
€8 be drawn parallel to two given straight lines to meet the
*ve, the rectangles of the segments will be to one another in
invariable ratio.
Let (#', y') be the given point and suppose a and/3 respec-
ely the inclinations of the given straight lines to the major
18 of the ellipse. By Art. 187 if a line be drawn from
, y) to meet the curve and be inclined at an angle a to the
ijor axis, the lengths of its segments are given by the
nation
182
RECTANGLE OF THE SEGMENTS OF A LINE.
r* (a* sin* a + V cos" a) +2r {f?\j sina + JVcosa)
therefore the rectangle of the segments = * . « tt—
° ° ar sin" a + J coi
Similarly the rectangle of the segments of the line
from (x>, y') at an angle 0- ^ffiy"^ -
Hence the ratio of the rectangles = 9 . o — —75 — r
a 2 sura + cos
this ratio is constant whatever x f and y' may be.
\0
Let be the point through which the lines OPp
are drawn inclined to the major axis of the ellipse at
a, )8, respectively ; then
OP. Op = a 2 sin'ft + V cos'ft
OQ.Oq~ a 2 sin 2 a + J 2 cos 2 a #
Draw the semi-diameters CD, CIS, parallel to 1
respectively, then, by Art. 206,
CZ> 2 _ a 2 sin 2 /3 + 5 2 co8 2 /3
CE* a 2 sin 2 a + i 2 cos 2 a ;
OP. Op CD*
• •
OQ.Oq CE 1 '
EXAMPLES ON THE ELLIPSE. 183
Let TM, TN be tangents parallel to Pp, Qq, respectively ;
sn if coincides with T, the rectangle OP. Op becomes
?lf 2 , and the rectangle OQ. Oq becomes TN 2 ;
TM 2 _CD 2 m
•'• TN 2 ~ CE 2 '
TM CD
• •
TN" CE*
EXAMPLES.
1. CP and CD are conjugate semi-diameters ; given the
►-ordinates of P (x', y), find the equation to the line PD.
2. If lines drawn through any point of an ellipse to the
Ktremities of any diameter meet the conjugate CD in the
^mts M, N, prove that CM. CN= CD".
3. CP, CD are two conjugate semi-diameters ; CP, CD'
De two other conjugate diameters ; shew that the area of the
rfangle PCP is equal to the area of the triangle DCD'.
4. Normals at Pand D, the extremities of semi-conjugate
iameters, meet in K; find the equation to KC 9 and shew that
ZG is perpendicular to PD.
5. In an ellipse the rectangle contained by the perpen-
icular from the centre upon the tangent, and the part of the
>rresponding normal intercepted between the axes is equal
> the difference of the squares of the semi-axes
6. Shew that the locus of the intersection of the perpen-
[cular from the centre on a tangent to the ellipse is the
irve which has for its equation r 8 = a 2 cos 2 + b 2 sin 2 0, the
jntre being the origin.
7. From A the vertex of an ellipse draw a line ARQ to
) the middle point of HP meeting SP in R ; shew that the
)cus of R is an ellipse, and also the locus of Q.
>4 EXAMPLES ON TIIE ELLIPSE.
8. Find the polar equation to the ellipse, the vertex Ik
the origin aod the major axis the initial line.
9. If any chord AQ meet the minor axis produced ii
and CPba a semi -diameter parallel to AQ, then
AQ.AR^lCP 1 .
10. A circle is described upon AA' the major axis o
ellipse as diameter ; P is any point in the circle ; AT.
are joined cutting the ellipse in points Q and Q' respecti 1
shew that
AP AT^ a' + b*
AQ + 2q b* ■
11. If circles be described on two semi-conjugate di
ters of an ellipse as diameters, the locus of their interse
is the curve defined by the equation
2 (s* + ff = flV + ty.
12. CP, CD are conjugate semi-diameters ; CQ is
pendicular to PD, find the locus of Q.
13. Find the points where the ellipse a (1-e*) =r+re
cuts the line a (1 - e") = r sin 6 + r (1 + e) cos 6.
14. Write down the polar equations to the four tan|
at the ends of the latera recta ; also the equations to the
gents at the ends of the minor axis ; the focus being the
15. Determine the locus of the intersection of tan,
drawn at two points P, Q, which are taken so that the
of the angles ASP, ASQ, is constant.
16. If PSp be a focal chord of an ellipse, and alon
line SP there be set off SQ a mean proportional betwee
and Sp, the locus of Q will be an ellipse having the
excentricity as the original ellipse.
17. Two ellipses have a common focus and their i
axes are equal in length and situated in the same sir
line; find the polar co-ordinates of the points of i
section.
EXAMPLES ON THE ELLIPSE. ' 185
S. From an external point two tangents are drawn to an
; between what limits does the ratio of the length of
gent to the other lie ?
S. TP, TQ are two tangents to an ellipse, and CP', CQ\
lie radii from the centre respectively parallel to these tan-
a, prove that F Q' is parallel to PQ.
>0. An ellipse and a circle cut in four points ; shew that
common chords make equal angles with the major axis of
ellipse.
81. When the angle between the radius vector from the
ip and the tangent is least, the radius vector = a.
12. When the angle between the radius vector from the
*e and the tangent is least, the radius vector = f — - — J .
S3. PT y pt are tangents at the extremities of any diameter
if an ellipse ; any other diameter meets PT in T, and its
agate meets pt in t ; also any tangent meets PT in T' and
& f; shew that PT: PT' :: pi :pt.
84. From the ends P, J), of conjugate diameters in an
«e, draw lines parallel to any tangent line ; and from the
ire C draw any line cutting tnese lines and the tangent in
ts p 9 d, t, respectively ; then will
Cp*+cd*=ce.
55. If tangents be drawn from different points of an ellipse
jngths equal to n times the semi-conjugate diameter at
point, then the locus of their extremities will be a con-
ic ellipse with semi-axes equal to
aVV+1) and b*J(n 2 +l).
16. Apply the equation to the tangent in Art. 171 to find
ocus of the intersection of tangents at the extremities of
igate diameters.
17. If from a point (#', y') of an ellipse a chord be drawn
Llel to a fixed line, shew that the length of this chord
EXAMPLES ON THE ELLIPSE.
where <f> is the inclination ol
tangent at (*', y) to the axis, and a the inclination of the
line to the axis,
28. If through any point P of an ellipse two chori
PR, be drawn parallel to two fixed lines and mtfciw i
a and 3 respectively with the tangent at P, shew thar il
PQ coaec a : PR cosec 8 is constant.
29. A parabola ia touched at the extremities of th(
rectum by an ellipse of given magnitude ; find the
rectum of the parabola.
30. The perpendicular from the centre on alinejoinii
ends of perpendicular diameters of an ellipse is of col
length.
31. Chords are drawn through the end of an axis
ellipse ; find the locus of their middle points.
82. Chords of an ellipse are drawn through anj
point ; find the locus of their middle points.
33. Two focal chords are drawn in an ellipse n
angles to each other ; find their position when the r«
contained by them has respectively it3 greatest am.
value.
34. In an ellipse if PP' and QQ" be focal chords t
angles to each other
SP.SP + BQ.BQ AO ,r BC tm
35. PSp, QSq, are focal chords ; suppose T th<
where the fines PQ, pq meet ; shew that rS is eijni
clined to the focal chords, and that T is on the d
"■ ; to 8.
36. If r, 6 be the polar co-ordinates of a point /
that
„ n „ h , 1+CCOS0
tan HPZ= -77- j — m and = ^-^~
V(2ar-r* — &') esm0
EXAMPLES ON THE ELLIPSE. 187
17. Perpendiculars are drawn from P and D the ex-
ities of any pair of conjugate diameters on the diameter
c tan a ; shew that the sum of the squares of the perpen-
iars is a 8 sin 2 a + i 8 cos 8 a.
18. The excentric angles of two points P and Q are <f> and
sspectively; shew that the area of the parallelogram
ed by the tangents at the extremities of the diameters
lgh P and Q is -s — j-p — it ; shew also that the area is
° sin (<p — <p)
when P and Q are the extremities of conjugate dia-
rs.
►9. Shew that the equation to the locus of the middle
ts of all chords of the same length (2c) in an ellipse is
v 4 8 • T4 2 ~ 2* 12 X— V.
a y + oixr a b
Q. Chords of an ellipse are drawn at right angles to one
her through a point whose co-ordinates are h, k; if
GQ be the radii drawn from the centre parallel to
chords, and E, F the middle points of tne chords,
' that
OE 2 OF^_tf Jf
CP* + CQ i ~a* + b*'
1. Given the co-ordinates of P, find those of the inter-
3n of the tangents at P and D. (See Fig. to Art. 192.)
2. Shew that the equation
£ . £_- 1 - \ x ( hx ' - a y') . y W + Sa?/ ) il a
a 8 + J a l ~\ a 2 b + ab* " l )
jsents the tangents at P and D, supposing x\ y' the co-
tates of P. (See Fig. to Art. 192.)
3. If GP, CD be any conjugate diameters of an ellipse
WA', and PP, BD be joined and also AD, A'P, these latter
secting in } shew that BDOP is a parallelogram.
EXAMPLES ON THE ELLIPSE.
44. Shew tliat the area of the parallelogram in tbe pre
ing question = ay' + bx' — ab, where x', y are the co-ordin
of P; and find the greatest -value of this area.
45. If a line he drawn from the focus of an ellipse to e
a given angle a. with the tangent, shew that the locus of
intersection with the tangent will be a circle which toncto
falls entirely without the ellipse according as cos a is Is
greater than the excentricity of the ellipse.
46. In an ellipse SQ, HQ, drawn perpendicular to »
of conjugate diameters, intersect in Q ; prove that the loci
Q is a concentric ellipse.
47. Two ellipses have their foci coincident ; a tango
one of them intersects at right angles a tangent to the o!
shew that the locus of the point of intersection is a t
having the same centre as the ellipses.
48. What is represented by the equation x* + y*=c'*
the axes are oblique?
49. Shew that when the ellipse is referred to any pa
conjugate diameters as axes, the condition that ^ = mi
c may represent conjugate diameters is mm' = — f;
50. The ellipse being referred to equal conjugate di
ters, find the equation to the normal at any point.
51. From any point P perpendiculars PM, P-V are <1
nn the equal conjugate diameters; shew that the n
bisects MN,
( 189 )
CHAPTER XL
THE HYPERBOLA.
. To find the equation to the hyperbola.
» hyperbola is the locua of a point which moves so that
ance from a fixed point bears a constant ratio to its
b from a fixed straight line, the ratio being greater
lity.
<T
W
; ITbe the fixed point, YY 1 the fixed straight line. Draw
rpendicular to YY'; take as the origin, OB" as the
>n of the axis of #, OY as that of the axis qf y .
; P be a point on the locus ; join HP, draw PM parallel
and PJVjparallel to OX. Let OH=jp, and let e be the
* HP to PN. Let x, y be the co-ordinates of P.
190 EQUATION TO THE HYPERBOLA.
By definition
J3P= ePN;
.-. HP*=e*PN*;
.\ PM* + HM* = <?PN\
that is, y 2 + (x —p)* = eV.
This is the equation to the hyperbola with the as
origin and axes.
210. To find where the hyperbola meets the axi
we put y = in the equation to the hyperbola; thus
(x — pY = e*a?;
•\ x — p = ±ex;
1 + e
Since 6 is greater than unity, 1 — e is a negative quanti
Let OA' = -±-— , 0-4 = -^-— the former being me
e — 1 1 +6 °
to the left of 0, then A' and -4 are points on the hyp
A and A' are called the vertices of the hyperbola, ;
the point midway between A and A' is called the ce
the hyperbola.
211. We shall obtain a simpler form of the equa
the hyperbola by transferring the origin to A or G.
I. Suppose the origin at A.
Since OA = „ , , we put x = x' + -±— ■ and sub
1 + 6 r 1 + e
this value in the equation
y* + (x -p)* = eV ;
thu8 ^ + ^_ p )W(*' + r ^)\
or
*+('-I?!)-'('*lf-.)'
EQUATION TO THE HYPERBOLA, 191
/. y a = 2pex' + (e 2 - 1) x ,%
!Hie distance A' A = -~r + t*— - fl « ; we will denote
6 — 1 1 + e e 2 - 1 '
"by 2a ; hence the equation becomes
tf = (e 2 - 1) (2oa>' + a;' 2 ).
\Ve may suppress the accent, if we remember that the
;in is at the vertex A, and thus write the equation ,
y = (e»- l) (2ax + x*) (1).
31. Suppose the origin at G.
Since GA = a, we put x = x' — a and substitute this value
Jl)_j thus
^=(e»-l){2a(a/-a) + (a'-a) 8 }
: = (e"-l)(a/ 2 -a s ).
We may suppress the accent, if we remember that the
gin is now at the centre (7, and thus write the equation
/ = (e« - 1) (a» - «») (2).
In (2) suppose x = 0, then y 2 = — (e 2 — 1) a 2 ; this gives an
possible^ value to y, and thus the curve does not cut the
m of y. We shall however denote (e 2 — 1) a 2 by 5 2 , and
asure off the ordinates GB and GB' each equal to b, as we
31 find these ordinates useful hereafter.
Thus (1) may be written
y> = b j (2ax + x*) (3),
1 (2) may be written
y* =£,(**-«') (4),
i
192 FORM OF THE HYPERBOLA.
or, mote symmetrically,
■3-fJ=»l, or, «y-JW. o*fi*..
1 + e
OA =
-1
1 + e e '
Cff= GA + AS- a + (e-l)a = ea,
CO=CA-OA = a -—<* = -,
e e
and OS=p = °-^^-.
213. We may now ascertain the form of the hyp
Take the equation referred to the centre as origin,
f--,(°?-<t)-
For every value of x less than a, y is impossible.
x = a, y = 0. For every value of x greater than a
It'
r
K.
B
?
J
w
E '^-^^'
B'
ta
.-"
FORM OP THE HYPERBOLA* 193
two values of y equal in magnitude but of opposite sign.
saice if P be a point in the curve on one side of the axis
«, there is a point P' on the other side of the axis, such
*t P'M=PM. Hence the curve is symmetrical with re-
«t to the axis of x, and it extends indefinitely to the right
If we ascribe to x any negative value we obtain for y
'* same pair of values as when we ascribed to x the cor-
ponding positive value. Hence the portion of the curve
"•he left of the axis of y is similar to the portion to the
Kit of it.
As the equation (1) may be put in the form
rf-jV+p) .-(*).
see that the axis of y also divides the curve symmetrically,
Eft that the curve extends above and below CA. Thus tne
gfre consists of two similar branches each extending inde*
*tely.
-The line .EST is the directrix, H is the corresponding focus,
the curve is symmetrical with respect to the line BOB',
flows that if we take 0£= GH and GE' = GE, and draw
' perpendicular to GE\ the point 8 and the line E'K'
form respectively a second focus and directrix, by means
^hich the curve might have been generated.
214. The point G is called the centre of the hyperbola,
csause every chord of the hyperbola which passes through C
Trisected in G. This is proved in the same manner as the
^responding proposition in the ellipse. (See Art. 163.)
215. We have drawn the curve concave towards the axis
& ; the following proposition will justify the figure.
The ordinate of any point of the curve which lies between
Vertex and a fixed point of the curve on the same branch
the vertex is greater than the corresponding ordinate of the
Faight line joining that vertex and the fixed point.
T. C. S. 13
196 TANGENT TO AN HYPERBOLA.
Hence SP— HP=2a; that is, the difference of the fool
distances of any point on the hyperbola is equal to the trot
verse axis.
219. The equation y* = -5 (a* — a 2 ) may be written
a
V % = is fc -a)(x + a).
Hence (see Fig, to Art. 213),
PM* _BC*
AM. AM' AC 1 '
Tangent and Normal to an Hyperbola.
220. To find the equation to the tangent at any poinlJj\
an hyperbola.
By a process similar to that in Art. 170, it will be fofflil
that the equation to the tangent at the point (a?', y') is
or a*yy ' — Vxx' = — a*b\
These equations may be derived from the corresponding
equations with respect to the ellipse by writing — ft 2 for i 1 .
221. The equation to the tangent to the hyperbola JDtJ
also be written in the form (see Art. 171)
y =5 mx + V(wV — J 8 ).
Conversely every line whose equation is of this form, is
tangent to the hyperbola.
222. It may be shewn as in the case of the circle that*
tangent to an hyperbola meets it in only one point. Also if*
. NORMAL TO AN HYPERBOLA. 197
e meet an hyperbola in only one point, it is in general
* tangent to the hyperbola at that point. For suppose
be the equation to an hyperbola, and
y =*mx + c
* equation to a straight line. Then to determine the
acissae of the points of intersection, we have the equation
a 2 (7wa; + c) 2 -iV = -a 2 i 2 ,
(aW - V) a? + 2a*mcx + a 2 (c 2 + b*) = 0.
This equation has always two roots, except
(1) when aWc 2 = (aW - b*) a 2 (c 2 + J 2 ),
d consequently the line is a tangent;
(2) when c?rr? — J 2 = 0; the equation then reduces to one of
B first degree, and therefore has but one root. Thus a line
lich meets the hyperbola in one point only is the tangent at
9t point unless the inclination of the line to the transverse
is be itan" 1 -.
223. The tangents at the vertices A and A' are parallel
the axis of y. (See Art. 172.)
224. To find the equation to the normal at any point of
* hyperbola. (See Art. 173.)
It will be found that the equation to the normal at
y
198
PROPERTIES OF THE HYPERBOLA*
This may also be written in the form
y = mx —
V(a*-£W)
. (See Art 174.)
225. We shall now deduce some properties of the hy
"bola from the preceding articles.
Let x, y' be the co-ordinates of P; let PTbe the tan
at P, PO the normal at P; PM, PN perpendiculars on
axes.
The equation to the tangent at P is
cfyy* — b*xx* = — c?b*.
Let y = 0, then x = -7 , hence
° CM'
Similarly
.-. CM.<JT=CA\
' CN.CT^CB*.
PROPERTIES OP THE HYPERBOLA, 199
226. As in Art. 176, we may shew that
CG = e*GM,
2 2
are
CG'^^-PM.
227. As in Art. 177, we may shew that
r{ * "IT 9 """J 5 "'
.«re &P=/, J5TP=r.
228. The tangent at any point bisects the angle between
t Jbcal distances of that point.
For in the manner given in Art. 178, we may shew that
SPG' = HPG;
A therefore since PT is perpendicular to GG\
TPS=TPH.
Or we may prove the result thus,
CG = eV (Art. 226) ;
•\ SG = e*x +ae,
HG = e V — ae.
Also SP=ex' + a, HP '= ex' — a; hence . .
SG _SP
HG~HP ;
icrefore by Euclid, vi. 3, PG bisects the angle between HP
id SP produced, that is,
SPG' = HPG.
229. To find the locus of the intersection of the tangent at
my point with the perpendicular on it from the focus.
It may be proved as in Art. 180, that the required locus is
ie circle described on the transverse axis as diameter.
200
TWO TANGENTS FROM AN EXTERNAL POINT.
230. Let p denote the perpendicular from E on
tangent at P, and p the perpendicular from S; then, as
Art. 181, it may be shewn that
Since
L2 »
. Vr „ 8V
.\ pp' = J*.
r = 2a + r, we have
p =
2a + r
231. From any external point two tangents can be
to an hyperbola.
Let A, k be the co-ordinates of the external point, then;
in Art. 182, we shall obtain the following equation for if*'
mining the abscissae of the points of contact of the tanj
and hyperbola,
The roots of this quadratic will be possible if
a W + a 4 (i 2 + V) (a% 2 - b*h*) is positive ;
that is, if
tfa*-b 2 h* + a*b*
is positive.
But if (A, lc) be an external point the last expressions
dtive, and therefore two tang
typerbola from an external point.
The product of the two values of x given by the abort
quadratic is
Eositive, and therefore two tangents can be drawn to
i - - -
a* If - ttK
X/,2 >
this product is therefore positive or negative according tf
a a & a — b*h* is negative or positive; that is, the two tangents
meet the same branch or different branches according «
a a &* — b*Jf is negative or positive.
INTERPRETATIONS OP AN EQUATION. 20X
The case in which a 2 & 9 — J*A 2 = requires to be noticed.
"« one root of the quadratic equation becomes infinite, and
other is ^ ,,,, ; see Algebra, Chapter XXII.
JtCL tit •
In this case the point (h, k) falls on a certain line called
asymptote, which we shall consider hereafter; see Art. 255.
* asymptote itself may then count as one of the two tan-
is from the point (h, ty. If A = and k = the point
X) is the origin; in this case the two asymptotes may
nt as the two tangents from the point (h, k).
232. Tangents are drawn to an hyperbola from a given
carnal point; to find the equation to the chord of contact.
h 9 h be the co-ordinates of the external point ; thtfn
equation to the chord of contact is
a*ky - b 2 hx = - a*b*. (See Art. 1 83.)
233. Through any fixed point chords are drawn to an
*erbola, and tangents to the hyperbola are drawn at the
* m emities of each chord; the locus of the intersection of the
gents is a straight line.
Let h, Jc he the co-ordinates of the point through which
* chords are drawn, then the equation to the locus of the
arsection of the tangents is
a'ky - b*hx = - a 2 b\ (See Art. 184.)
234. If from any point in a straight line a pair of tan-
it* be drawn to an hyperbola, the chords of contact will all
ft* through a fixed point. (See Art. 185.)
235. The student should observe the different interpre-
ions that can be assigned to the equation
a*ky — Wfcc = — a a i 9 .
The statements in Art. 103 with respect to the circle may
be applied to the hyperbola.
EXAMPLES ON THE HYPERBOLA,
I
1. " "What i3 the equation to an hyperbola of given h
verse axis whose vertex bisects the distance between
centre and focus ?
If the ordinate MP of an hyperbola be produced
so that M Q = SP, find the locus of Q.
3. Any chord AP through the vertex of an hyperbt
divided in Q so that AQ : QP :: AC : BC, and Q
drawn to the foot of the ordinate MP; from Q a line is d
at right angles to QM meeting the transverse axis in 0;
thztAOiA'O-.-.AC-.BG 1 .
4. PQ is a chord of an ellipse at right angles to thi
jor axis AA'; PA, QA' are produced to meet in It; shew
the locus of 11 is an hyperbola having the same axes a
ellipse.
If a circle be described passing through any po
of a given hyperbola and the extremities of the trsne
axis, and the ordinate MP be produced to meet the cin
Q, shew that the locus of Q is an hyperbola whose conj
axis is a third proportional to the conjugate and tram
axes of the original hyperbola.
6. Find the locus of a point such that if from It a p
tangents be drawn to an ellipse the product of the pe
diculars dropped from the foci upon the chord of contac
be constant.
( 203 )
CHAPTER XII.
THE HTPEBBOLA CONTINUED.
Diameters.
236. To find the length of a line dratonfrom any point in
direction to meet an hyperbola.
"Get x\ y' be the co-ordinates of the point from which the
* is drawn; x, y the co-ordinates of the point to which the
is drawn ; 6 the inclination of the line to the axis of x ;
ue length of the line ; then (Art. 27)
x = x' + rco&0 9 y = y' + rsin0.... ........ (1).
If (a?, y) be on the hyperbola these values may be substi-
*A in the equation a 9 y 9 — b V = — c?b* ; thus
a 9 {y' + r sintf) 9 - b* {x' + r cos d)* = -a*b*;
f(a 1 Bin ,l d-i 9 cos 9 d) + 2r(aysin^-JVcos^)
+ ay 9 -JV 9 + a 9 J 9 = (2).
Prom this quadratic two values of r can be found which
the lengths of the two lines that can be drawn from (x\ y)
Old given direction to the hyperbola.
237. To find the diameter of a given system of parallel
rds in an hyperbola. (See definition in Art. 148.)
Let be the inclination of the chords to the transverse axis
the hyperbola ; let x, y be the co-ordinates of the middle
lit of any one of the chords ; the equation which deter-
ies the lengths of the lines drawn from (&', y) to the curve
[Art. 236)
vfBm 9 0-V(^^6) + 2r(a i y , smd^b % x r co&0)
+ ay 9 -iV 9 + a 9 J 9 = (1).
204 CONJUGATE DIAMETERS OP THE 1ITPEBBOLJ
Since (x, y) is the middle point of the chord, the valw
r furnished by this equation must be equal in magnituilt
opposite in sign ; hence the coefficient of r must vanish ; I
a*y sin 8 — Wx C08 6 = 0, or y'=-;cot0 . x (3).
Considering x' and y as variable this is the equation It
straight line passing through the origin, that is, througln
centre of the hyperbola.
Hence every diameter passes through the centre.
Also every straight line passing through the centres
diameter, that is, bisects some system of parallel chords, I
by giving to a suitable value the equation (2) may be mi
to represent any line passing through the centre. If f
the inclination to the axis of x of the diameter which
all the chords inclined at an angle 6, we have from (2)
238. If one diameter bisect all chords parallel to asu
diameter, the second diameter will bisect all chords pan
to the first.
Let &j and 6 7 be the respective inclinations of the i
diameters to the transverse axis of the hyperbola. Since
first bisects all the chords parallel to the second, we hare
tan 8. tan 6, = —..
■ * a
And this is also the only condition that must hold in wis
that the second may bisect the chords parallel to the first.
The definition in Art. 191 holds for the hyperbola.
239. Every straight line passing through the centre of*
ellipse meets that ellipse ; this is evident from the figure,*
it may be proved analytically. But in the case of an hyp*
bola this proposition is not true, as we proceed to shew.
CONJUGATE HTPEEBOLA. 205
240. To find the points of intersection of an hyperbola
:* a straight line passing through its centre.
the equation to the straight line be
y = mx.
^Substitute this value of y in the equation to the hyperbola
Q we have for determining the abscissae of the points of
mection the equation
a 9 i 9
• » OCT =s
— a m
Hence the values of x are impossible if aW is greater
Thus a line drawn through the centre of an hyperbola will
fc meet the curve if it makes with the transverse axis on
her side of it an angle greater than tan" 1 - .
241. It is convenient for the sake of enunciating many
jperties of the hyperbola to introduce the following im-
ttant definition.
Def. The conjugate hyperbola is an hyperbola having
1 its transverse and conjugate axes the conjugate and trans-
ne axes of the original hyperbola respectively.
242.- To find the equation to the hyperbola conjugate to a
sen hyperbola.
Let AA' t BB be the transverse and conjugate axes respec-
fldy of the given hyperbola; then BB is the transverse
5i of the conjugate Hyperbola, and A A is its conjugate
OS. Let P be a point in the given hyperbola, Q a point in
« conjugate hyperbola. Draw PM 7 QN perpendicular to
206
EQUATION TO THE CONJUGATE HYPERBOLA*
OX, CY respectively. The equation to the given hyp
is
a"
Hence
since Q is a point on an hyperbola having CB, CA i
semi-transverse and semi-conjugate axes respectively,
if x f y denote the co-ordinates of Q,
a
rf-yfc'-?)-
This, therefore, is the equation to the conjugate hype
we observe that it may be deduced from the equation
given hyperbola by writing — a* for a* and — J 1 for V.
STRAIGHT LINES THROUGH THE CENTRE. 207
The foci of the conjugate hyperbola will be on the line
'!¥ at a distance from u = AB (Art. 216) ; that is, at the
te distance from G as 8 and H.
243. Every straight line dravm through the centre of an
>erbola meets the hyperbola or the conjugate hyperbola, except
two lines inclined to the transverse axis of the hyperbola
xn angle =* tan" 1 - .
Xiet the equation to the straight line be
y = mx (1).
To find the abscissas of the points of intersection of (1)
h. the given hyperbola, we nave, as in Art. 240, the
Lation
^-y-aW W *
Similarly to find the points of intersection of (1) with the
rjugate hyperbola, we have the equation
21*
a*b
& — "a q 75 ••• • • (3).
am —o v '
V
If m* be less than -5 , (2) gives possible values, and (3)
a
possible values for x ; if m 9 be greater than — a , (2) gives
a
b*
possible values, and (3) possible values for x\ if m* = -j ,
a
and (3) make x infinite. Thus the two lines that can
drawn at an inclination tan" 1 - to the transverse axis of
a
s given hyperbola meet neither curve ; and every other line
ets one of the curves.
244. Of two conjugate diameters one meets the original
Terbola, and the other the conjugate hyperbola.
Let the equations to the two diameters be
y = mx, y' = m'x;
CONJIOATE HYI'EKBOLAS.
then, by Ait. 238,
the first diameter meets the original hyperbola, and lb
second the conjugate hyperbola. If m 1 is greater l'
-5 , m" is teas than -5 ; thus the first diameter meets
a a
conjugate hyperbola, and the second the original hypeiboli
245. We proceed now to some properties connected
conjugate diameters. When we speak of the extremities of
diameter we mean the points where that diameter inters'
the original hyperbola or the conjugate hyperbola.
We may remark that the original hyperbola bears
same relation to the conjugate hyperbola as the conjuo
hyperbola bears to the original hyperbola. Thu3 the den
tion may be given as follows : two Hyperbolas are called
jugate when each has for its transverse axi3 the conjngil
axis of the other.
Also if a line bisect all parallel chords terminated n
of the hyperbolas it bisects all the chords of the same ays'
which are terminated by the other hyperbola. For the ep*
tion (Art. 237) tan 6 tan & = — remains unchanged when «
write — a ! for a* and —V for & 5 , that is, when we pass I
the original hyperbola to the conjugate (Art. 242).
Both curves are comprised in the equation
(«y- sv) 1 -«'»'•
240. Tfie tangent at either extremity of any (ZJOfUttt
parallel to the chorda which lira! diameter bisects. See Art.l3*l>
247. Given the co-ordinates of one extremity of a iiaiaOM
to find those of either extremity of the conjugate
Let AC A, BCB be the axes of an hyperbola; POP
CONJUGATE DIAMETEES OF THE HTPEEBOLA.
209
(BCD' a pair of conjugate diameters. Let x\ y be the
Ktaa co-ordinates of P; then the equation to CJP is
is
= 1
y = *-,x
x
a).
Knee the conjugate diameter DD' is parallel to the tangent
►P, the equation to Z>Z>' is
* = w x (2) -
We must combine (2) with the equation to the conjugate
Sperbola to find the co-ordinates of D and D\ Substitute
*om (2) in
ay-JV = aW; then
b A x n
a y
•• a*P~ b* '
.'. x = ±
b '
T.C.8.
14
CONJUGATE DIAMETERS OF THE HTPEEi
. from (2), y = ± — .
mil duid
In the figure the abscissa of D is positive, and tl
negative ; hence the upper sign applies to D, and the lm
toiy.
of the squares of two conjugate m
The differen
is constant.
Let a
article,
' be the co-ordinatea of P; then, by the precedi
b'x'-d
!.|
Hence the difference of the squares of two conjugate «
diameters is equal to the difference of the squares of thewi
axes.
240. The area of the parall loyrani funned hi/ tanged
the ends of conjugate diameters is constant.
Let PCP, BCD' be the conjugate diameters (see Fig.
Art. 247). The area of the parallelogram formed by tanga
at P, D, F, D', is 4CP. GDsmPCD, or 4p. CD, when
denotes the perpendicular from C on the tangent ot P. I
x\ y' be the co-ordinates of P; then the equation to I
tangent at P is
= JV &
V <fy x if
Hence (Art. 47)
PEEPENDICULAE FROM THE CENTEE ON THE TANGENT. 211
.\ 4p.GB = 4ab.
Hence the area of any parallelogram formed by tangents at
e ends of conjugate diameters is equal to the area of the
ctangle formed by tangents at the ends of the axes.
250. Let a\ V denote the lengths of two conjugate semi-
imeters; a the angle between them; by the preceding
tide,
dV sin a = a5.
By making P move along the hyperbola from A we can
ie GP or a as great as we please. Also since a' 8 — J' 2 is
istant, V increases with a. Thus sin a can be made as
all as we please, that is, GP and CD can be brought as
*r to coincidence as we please. The limiting position
Fards which they tend is easily found; for from Art. 237,
mm =•-•;
a
is the limit to which m and m' approach as CP and CD
•L
Broach to coincidence is + - .
• a
251. From Art. 249 we have - •
212 -212
^' CP-^y (Art. 248.)
This gives a relation between p the perpendicular from the
ltre on the tangent at any point P 9 and the distance GP of
it point from the centre.
Also if <f> denote the angle which the perpendicular makes
th the transverse axis, we may shew as in Art. 196 that
14-*
212 EQUATION TO THE HYPERBOLA REFEKKED
252. To find the equation to the hyperbola referred M
pair of conjugate diameters as axes.
Let CP, CD be two conjugate semidiameters (see Fig. H
Art. 247), take CPas the new axis of x, CD ns that of j;
tetPCA=a, DCA=fi. Let x, y be the co-ordinates of
any point of the hyperbola referred to the original axes;
x', y the co-ordinates of the same point referred to the ict
axes ; then (Art. 84}
x = x' cos cl-t y' cos j3,
y^x'sma+y'amp.
Substitute these values in the equation
a*y x — b*x* = — a*b* ;
then a*(x'Bma+y'smfi)*— i* (#' cos a + y cos #)'
or x* (a* sin' a — J* cos" a) +y* (a'sin'/S — i*cos 5 /3)
+ 2x'y' (a* sin a sin & — V cos a cos /9) = — a*b\
But since CPand CD are conjugate semidiameters,
tancttanjS=- 1 :
a
hence the coefficient of x'y' vanishes, and the equation be-
comes
as* (a* sin 1 a — o* C03* a) -f y " (a 1 sin* £ - b' cos* y3) = - JS<
In this equation suppose y = 0, then
a* sin* a — b' cos' a 5* cos* a — a' sin* a '
This is the value of CP' which we shall denote by a".
we put x' = in the above equation, we obtain
-aV
TO CONJUGATE DIAMETERS AS AXES. 213
w since we have supposed that the new axis of x meets
rve, we know that the new axis of y will not meet the
(Art 244), so that
cffan'fi-b'caPfi
a positive quantity ; we shall denote it by — V*. Hence
uation to the hyperbola referred to conjugate dia-
ls
a" b" ~ '
ipressing the accents on the variables,
a* &'* ~ l *
30 the equation to the conjugate hyperbola referred to
ne axes is
a b
e equation to the tangent to the hyperbola will be of the
farm whether the axes be rectangular or the oblique
l formed by a pair of conjugate diameters. (See Art.
J. Tangents at the extremities of any chord of an hyper-
eet in the diameter which bisects that chord. (See Art.
L If a chord and diameter of an hyperbola are parallel,
yplemental chord is parallel to the conjugate diameter,
xte. 202, 203.)
Asymptotes.
5. The properties of the hyperbola hitherto given have
tmilar to those of the ellipse ; we have now to consider
nroperties peculiar to the hyperbola.
214
ASYMPTOTES.
Let the equation to the hyperbola be
and let GL be the line which has for its equation
hx
9 a
Let MPQ be an ordinate meeting the hyperbola in
the straight line GL in Q ; then if CM be denoted by t
If then the line MPQ be supposed to move parallel t
from A, the distance PQ continually diminishes, and by
ASYMPTOTES. 215
large enough we may make PQ as small as we please.
«e line GL is called an asymptote of the curve.
Similarly the line GL\ which has for its equation
bx
asymptote.
Thus the equation
a o
tcdudes Txrfih asymptotes. We may take the following de-
■dfion.
~ Def. An asymptote is a straight line the distance of
irich from a point of a curve diminishes without limit as
*• point in the curve moves to an infinite distance from
*« origin.
The distance of P from GL is PQ sinP#(7; and as we
^Ve seen that PQ diminishes without limit as P moves away
*tol the origin, GL is an asymptote according to the definition
**e given.
256. In the same manner we may shew that GL is an
fcymptote to the conjugate hyperbola. For let MP be pro-
ceed to meet the conjugate hyperbola in P', then (Art. 242)
... P'Q = ly(a* + a*)-x}= *" -
a 1 ■ ' ) */(ar + a)+x
Hence as CM is increased indefinitely P'Q is diminished
indefinitely; therefore GL is an asymptote to the conjugate
hyperbola.
257. The equation to the tangent to the hyperbola at Jhe
point («', y*) is
CONNEXION OF TANGENT AND ASYMPTOTE.
a*yy' — b'xx' = — o*J',
vc^'-a 1 } y
=7RT
If x and y are increased indefinitely the limiting fon
which the above equation approaches is
bx
Thus the tangent to the hyperbola approaches contini
to coincidence with an asymptote when the point of coi
s away indefinitely from the origin.
258. It appears from Art. 243 that every straight
drawn through the centre of an hyperbola mu3t meet
hyperbola or its conjugate, unless its direction coincides
that of one of the asymptotes. And from Art. 250 it apj
tthat as conjugate diameters increase indefinitely they appr
to coincidence with one of the asymptotes.
r
The line joining the ends of conjugate (Harnett
parallel to one asymptote <t»d bisected by the other.
Let &',y' be the co-ordinates of any point P on the hi
bola (see Fig. to Art. 247) ; then the co-ordinates of L
extremity of the conjugate diameter arc (Art. 247)
b a
Hence the equation to DP is
PROPERTIES OF THE ASYMPTOTES. 217
.tis, y _y=__ (x-x');
therefore DP is parallel to the asymptote
bx
* a
Also the co-ordinates of the middle point of DP are
. 10)
!(,+*).» |(, + S).
ay' + bx' , ay' + bx'
■tf is, -2—7 — and -^—^ .
pThese co-ordinates satisfy the equation
bx
re the asymptote y = — bisects Pi?.
Since the diagonals of a parallelogram bisect each other,
PD is one diagonal of tne parallelogram of which GP
CD are adjacent sides, the other diagonal coincides with
asymptote, that is, the tangents at P and D meet on the
frmptote.
r
260. The equation to the hyperbola referred to conjugate
iuxieters as axes is
Hence the equations to the asymptotes referred to these
9K68 are
b'x Vx , oN
218
EQUATION TO THE HYPERBOLA
For we may shew as in Art. 243 that the lines denoted
(2) are the only lines through the centre which meet neit
(1) nor its conjugate. Hence these lines are the asympfc
by Art. 258.
Or the aarne conclusion may he obtained thus; the orig
equation to the hyperbola is
and that to the two asymptotes
If by substituting for x and y their values in terras o!
new co-ordinates x' and y, and suppressing i
variables, the former equation is reduced to
the latter must become, by the same substitution,
261. To find the rtjimtion to tha hyperbola »
asymptotes as axes.
Let OX, CYbe the original axes; CX\ OT' the
axes, so that CX' and OY' are inclined to CX on Op;
sides of it at an angle a such that tan a = - - Let x,
the co-ordinates of a point P referred to the old axes;
the co-ordinates of the same point referred to the new
Draw PM' parallel to OY', and PM and M'N each pi
to CY. Then
x = CM=CN+NM
= (x 4- y) cos a.
REFEEBED TO THE ASYMPTOTES AS AXES. 219
•
y = PM = (y' — x') sin a.
a
the equation
-77-5 — 75T : substitute these
, , a' + J*
•jr 4-.
easing the accents,
a' + b*
xy j-.
equation to the conjugate hyperbola referred to the
a is (Art. 242)
ay=-
a 2 +»
!0 EQUATION TO TUE TANGENT.
262. To find the equation to the tangent at any point $
hyperbola when the curve is referred to its asymptotes at at
Let x\ y' be the co-ordinates of the point ;
%", y" the co-ordinates of an adjacent point on the ct
The equation to the secant through these points is
y-y - ^.,_^, («-«)...
Since (x f , y') and (x", y") are points on the h y
ay = i (<*'+**);
/. x"y" = x'y'.
Hence (1) may be written
y-y^-Jifc-aO.
Now in the limit x" =x'\ hence the equation to t
gent at the point [x', y') is
y-y' = -K(x-x')
This equation may be simplified; multiply by x', tin
. „ . , <* + P
yrt-rxy =2xy = — — .
263. To find where the tangent at (x' t y') meets
of x put y = in the equation
_ a* + 5' _ Zx'y __
POLAR EQUATION TO THE HYPERBOLA. 221
Similarly to find where the tangent cuts the axis of y put
* in the equation ; thus
y ~ 2x' "" x* ~* y -
Thus the product of the intercepts = ix'y* = a 9 + b\ The
m of the triangle contained between the tangent at any
3nt and the asymptotes is equal to the product of the
*pts into half the sine of the included angle
= £(a* + i") sin2a = (a 2 + S a ) sina cosa = aJ,
3 is therefore constant.
Since the tangent at (x\y r ) cuts" off intercepts 2a?', 2y\ from
It axes of x and y respectively, the portion of the tangent at
|r point intercepted between the asymptotes is bisected at
9 point of contact.
Polar Equation.
264. To find the polar equation to the hyperbola, the
being the pole.
Let HP=r, AHP=0; (see Fig. to Art. 209) ;
mi 3P= ePNy by definition ;
« is, EP=e{OH+HM);
a
r = a (e?-l) + er cos (tr-0), (Art. 212) ;
,\ r (1 + e cos 6) = a (e?— 1),
** r -l+ecos0 * W*
If we denote the angle XHP by 0, then we hare as before
EP=e{OH+EM);
222
thus
and
;POLAB EQUATION TO THE HYPEBBOLA.
r =
r =
a(e s — 1) +6r cos0,
a(e 2 -l)
1 — e cos
We may also proceed thus; in the figure to A
suppose SP=r ana PSH= :
then
that is,
or
and
SP=ePN',
SP=e{SM-SE');
r — er cos0 — a(e 2 — 1) ;
•\ r (e cos — 1) = a (e 2 — 1),
e cos — 1
265. It will be a good exercise to trace the form
hyperbola from any of these polar equations. Take 1
ample the equation (1); suppose = 0, then r=a|
we must therefore measure off the length a(e — 1)
initial line from the pole H, and thus obtain the poir
one of the points of the curve.
TRACING THE HYPERBOLA. 223
As increases from to — we see from (1) that r increases ;
\ is negative when is greater than — and r continues to
rease. Let d be such an angle that 1 + 6 cos a = 0, that is,
a =± — - , then the nearer approaches to a the greater r
somes, and by taking near enough to a, we may make r
great as we please. Thus as increases from to a that
tion of the curve is traced out which begins at A and
ises on through P to an indefinite distance from the origin.
When is greater than a, r is negative, and is at first in-
initely great and diminishes as increases from a to ir.
ice r is negative we measure it in the direction opposite tQ
it we should use if it were positive. Thus as increases
m a to 7T that portion of the curve is traced out which
gins at an indefinite distance from G in the lower left-hand
adrant, and passes on through Q to A'. HA' is found by
tting = 7T m (1) ; then r becomes — a (e + 1), therefore
4! is in length = a (e + 1).
As increases from ir to 2tt — a, r continues negative and
Dderically increases, and may be made as great as we please
F taking sufficiently near to 2v — a. Thus the branch of
e curve is traced out which begins at A! and passes on
rough Q to an indefinite distance.
As increases from 2ir — a to 2tt, r is again positive, and
at first indefinitely great and then diminishes. Thus the
ation of the curve is traced out which begins at an indefi-
tely great distance from G in the lower right-hand quadrant
A passes on through F to A.
The asymptotes GL and Gil are inclined to the transverse
is at an angle of which the tangent is - ; hence cos LGA
-77-5 — t5v=-, and cos Z C4' = ; that is, LGA' — a.
>/(ar+o 2 ) e' e'
Us as approaches the value a the radius vector approaches
a position parallel to CL. Similarly as approaches the
He 27r — a tne radius vector approaches to a position parallel
QL\
EBOLA.
224 EQUILATERAL OK RECTANGULAR HYPE]
266. As in Art. 205 it may be shewn that the
equation to a chord subtending at the focus an angle 2j£
e cos + sec /3 cos {a — ff) '
il a + ,S being respectively the vectorial angles i
ich join the focus to the ends of the chord, and
lines i
semi-la t us rectum,
Hence the polar equation to the tangent is
I
267. The polar equation to the hyperbola,
being the pole, is (Art. 206)
t* («' sin' 6 - b 1 cob' 8) = — a%'
Arts. 207, 208 are applicable to the Hyperbo]
the i
I
^Equilateral or Rectangular Hyperbola,
268. If in the equation to the ellipse a*y*+JV=
we BUppose b = o, we obtain a? + y 3 = a 1 , which is the eqn
to a circle ; so that the circle may he considered a parti
case of the ellipse. If in the equation to the hype
a*j? — b*a?=—c?f? we suppose J=a, we have y* — & =
We thus obtain an hyperbola which is called the emiSt
hyperbola from the equality of the axes. Since the i
between the asymptotes, which = 2 tan -1 - , becomes a
angle when b = a, the equilateral hyperbola is also calle
rectangular hyperbola.
The peculiar properties of the rectangular hyperboli
be deduced from those of the ordinary hyperbola by nu
Thus since l'= a' («*— 1) we have e" — 1 = 1, .*. i
The equation to the tangent is (Art. 220)
yy' -xx=-a*.
EXAMPLES ON THE HYPEBBOLA. 225
From Art. 227 PG=PG' = VK) •
The equation to the conjugate hyperbola is, by Art. 242,
y — x = a.
^ Thus the conjugate hyperbola is the same curve as the
ciginal hyperbola, though differently situated.
By Art. 248, CP= CD, and therefore by Art. 259, CP
ad UD are equally inclined to the asymptotes.
EXAMPLES.
1. The radius of a circle which touches an hyperbola and
B asymptotes is equal to that part of the latus rectum which
• intercepted between the curve and asymptote.
2. A line drawn through one of the vertices of an hyper-
ola and terminated by two lines drawn through the other
Brtex parallel to the asymptotes will be bisected at the other
lilt where it cuts the hyperbola.
3. If a straight line be drawn from the focus of an hyper-
fct the part intercepted between the curve and the asymptote
^ ; — 7i , where and a are the angles made respectively
Bin a + sin ° r J
' the straight line and asymptote with the axis.
4. PQ is one of a series of chords inclined at a constant
gle to the diameter AB of a circle, find the locus of the
Jit of intersection of AP and BQ.
5. Pis a point in a branch of an hyperbola, P is a point
Bt branch of its conjugate, CP, CP', being conjugate semi-
tneters. If S, 8' be the interior foci of the two branches,
>ve that the difference of SP and S'P' is equal to the dif-
SBce of AC and BC.
6. If x, y be co-ordinates of any point of an hyperbola,
rw that we may assume x = a sec 6, y = b tan 0.
T. c. & 15
EXAMPLES ON THE HYFEEBOLA.
/ meeting ti
7. A line is drawn parallel to the axis of ij meeting ti
hyperbola L i — rj = l, and its conjugate, in points P, Q;ssx
that the normals at P and Q intersect each other on the a
of x. Shew also that the tangents at P and Q intersect i
the curve whose equation is y* (a'y 1 — b V) — 4 JV.
8. Tangents to an hyperbola are drawn from any point
one of the branches of the conjugate ; shew that the chori
contact will touch the other branch of the conjugate.
Find the equation to the radii from the centre to the poia
of contact of the two tangents, and if these radii are perpd
dicular to one another, shew that the co-ordinates of the psil
from which the tangents are drawn are
9. Two tangents to a parabola include an angle a;
that the locus of their point of intersection is an hyperfo
with the same focus and directrix.
10. Under what limitation is the proposition inEsanf
30 of Chapter x. true for the hyperbola?
11. The ratio of the sines of the angles made byadisiri
of an hyperbola with the asymptotes is equal to the rate'
the sines of the angles made by the conjugate diameter.
12. With two conjugate diameters of an ellipse as asp!
totes a pair of conjugate hyperbolas is constructed; proved
if one hyperbola touch the ellipse the other will do so tl
wise ; prove also that the diameters drawn through the
of contact are conjugate to each other.
( 227 )
CHAPTER XIII.
GENERAL EQUATION OP THE SECOND DEGREE.
269. We shall now shew that every locus represented
f an equation of the second degree is one of those which
e have already discussed, that is, is one of the following ;
point, a straight line, two straight lines, a circle, a parabola,
ellipse, or an hyperbola.
The general equation of the second degree may be written
aa? + bxy + cy* + dx + ey +/=0;
shall suppose the axes rectangular; if the axes were
lique we might transform the equation to one referred to
tangular axes, and as such a transformation cannot affect
i degree of the equation (Art. 87), the transformed equation
II still be of the form given above.
If the curve passes through the origin /= ; if the curve
28 not pass through the origin /is not =0, we may there-
d divide by/and thus the equation will take the form
a'a? + b'xy + cY + d'x + e'y + 1 = 0.
270. We shall first investigate the possibility of removing
m the equation the terms involving the first power of the
riables.
Transfer the origin of co-ordinates to the point (A, h) by
tting
x=*(xf + h f y = y' + Jc,
1 substituting these values of x and y in the equation
aof + bxy + cif+dx + ey+f=0 (1);
15—2
223 <t:ntt..u. cuuyes of the second degree.
thus we obtain
ax" + hx'y' + cy" + [iah + bk + d) x + (2cfc + bh + e)y
+f = (2),
where /' = ah 3 + bkk + ctf + dh + eh +/. (3). '
Now, if possible, let such values be assigned to Ji and/; a
will make the coefficients of x and y vanish ; that is, let
2ah + bk + d = 0, and 2ch+bh + e = 0;
,. , 2cd— be , %ae—bd
thus A = is — ; — , * = -n — - — .
o — iac o — iac
It will therefore be possible to assign suitable
and k, provided b* — iac be not = 0.
We shall see that the loci represented by the general eqn
tion of the second degree may be separated into two cliaa
those which have a centre, and those which in general hii l-
nol a centre, and that in the former case i 5 — lac is ixn :.■■■■
and in the latter case it is zero. We shall first consider t)
case in which V~iac is not zero, and consequently tlievak*
found above for h and k are finite.
Equation (2) thus becomes
ax K +bx'y' + cy"+f' = (4),
Now if (4) is satisfied by any values se„ y, of the variably
it i.9 also satisfied by the values — x v —y r Tlence the w*
origin of co-ordinates is the centre of the locus represent!
\>y (i).
Thus if l* — iac be not =0, the locus represented by l'l
lias a centre, and its co-ordinates arc h and k, the valuer
which are given above.
The value of/" may be found by substituting the value**
h and h in (3) ; the process may be faeilitated thus ; wu h-
2ah + b?c + d=0,
i u .
TRANSFORMATION OP THE EQUATION. 229
lltiply the first of these equations by A, and the second
md add ; thus
2aA 2 + 2c#* + 2bkh + dh + eh = 0,
2/'-dA-e&-2/=0;
e shall retain f for shortness,
1. We may suppress the accents on the variables in
Dn (4) of the preceding article and write it
aa? + bxy + ctf+f = (5).
ds equation we shall further simplify by changing the
ons of the axes. (Art. 81.)
.t x=*x'co&0 — y' sin0,.
y = x' sin + y' cos 0,
ibstitute in (5) ; thus
os* + c sin" + b sin cos 0)
+ y 2 (a sin 2 + c cos 2 — b sin cos 0)
{2 (<?-«) sin 0cos 0+ b (cos 2 - sin 2 0)} +/* = 0...(6).
[uate the coefficient of xy' to zero ; thus
2 (c — a) sin cos + J (cos 2 - sin 2 0) = 0,
(c-a)sin20 + 5cos20 = O;
.-. tan20 = -^- (7).
nee can always be found so as to satisfy (7), the term
ing x'y 9 can be removed from (6), and the equation
es
EEE.
230 CENTRAL CURVES OF THE SECOND DEGREE.
a;' 5 [a cos s 6 + c sin 5 8 + b sin 6 cos 6)
+ y" (a sin 1 + c cos 5 8 — b sin 8 cos 8} +/'=
or Ax* + By'*+f = (s)
where A = £ {a + c + (a - c) cos 2$ + b sin 20],
B = £ [a + c - (a - c) cos 28 - b Bin 20}.
6
Since
tan 28 = -
sin20
V{S'+(«-c)r
V{&*+ (a- <=)V
Hence ^ = £[« + c + VI s ' + (a - «)')].
5 = £[« + c-V[6 5 +(«-c) , j].
We may suppress trie accents on the variables in (8) »
write it
A.B..
"7 J' y
(1) If A, B, and/' have the same sign, the locus isi
possible.
(2) If A and B have the same sign and f have the «
trary sign, the locus is an ellipse of which the semi-axesi
respectively
■1 and y/(-£. (Art. 160.)
The locus is of course a circle if A=B.
A-
(3) If A and B have different signs, the locus is
hyperbola. (Art. 211.)
We have supposed in these three cases that/' is not =
if/' = 0, and A and B have the same sign the locus is'
CENTRAL CURVES OF THE SECOND DEGREE. 231
igin; if/"=0, and A and B have different signs the locus
insists of two straight lines represented by
'-^(-b)*'
From the values of A and B we see that
AB Ja + cY-h>-(a-c)*
_ A.ac - V
Hence A and B have the same sign or different signs
©cording as 6* — 4ac is negative or positive.
272. Hence we have the following summary of the results
f the preceding articles of this chapter. The equation
ax* + hxy + cy* +dx + ey +/=
presents an ellipse if V — 4mc be negative, subject to three
tceptions in which it represents respectively a circle, a point,
fcd an impossible locus. If b* — 4ac be positive, the equation
presents an hyperbola subject to one exception when it
presents two intersecting straight lines.
273. We may notice that the equation in Art. 271,
b
tan 25 =
a—c*
is an infinite number of solutions ; for if 2a be one value of
' which satisfies the equation, then if 20 = 2a + titt, where n
any integer, the equation will be satisfied. But these dif-
tent solutions will all give the same position for the axes.
or the values of are comprised in the expression a +-5- *
*d by ascribing different values to n we obtain a series of angles
fcch differing from a by a multiple of — , and the only changes
lat will arise from selecting different values of n are that the
fcis of x in one case may occupy the position of the axis of y
232 CUKVES OF THE SECOND DEGREE WITHOUT A CESTEE.
in another and vice versa, or the positive and negative
tions of the axes may be interchanged.
The radical in the value of cos 20 and of sin 20 in Art. ffl
may have either sign ; but the sign must be the same in butft
in ordeT that the relation tan 20 = - may hold.
a — c "
274. It appears from the former part of Art. 271, thatbj
turning the axes through an angle 6 the equation
ax' + hxv + cf+f'^O
becomes ax' 1 + b'x'y + c'y' 1 +f — 0,
where a = ^ [a + c + (a - c) co&2& + b Bhi20],
V = (c- o) sin 2$ + i cos 26,
e' »± {a + c- (a-o) cos 28- b sin 28).
Hence a +c' = a + c; and
b'' — ia'a'= {{o — a) sin 20 + b cos 20}*
- (a + c)' + [{a - c) cos 2d + 1 sin 5
= (a- c y + b*-(a + cy
— b 1 — iac.
Thus the expression b 1 — \ac has the same value w!
be formed from the coefficients of the general erjuatii
second degree before or after the axes have been shifted.
The same remark applies to the expression a + c.
Hence we conclude that if the curve represented by d
general equation
ax* + bxy + cy* + dx + ey +/=*
be a rectangular hyperbola, a + c = ; for if the curve TO
referred to its transverse and conjugate diameters as ;-\
relation would hold, and therefore, as we have jusi
must always hold whatever be the axes.
275. We have next to consider the case in which
V - 4nc = 0.
«UBVES OF THE SECOND DEGREE WITHOUT A CENTRE. 233
We cannot now as in Art. 270 remove the terms involving
^ first power of the variables from the general equation,
fc we can still simplify the equation as in Art. 271, by
mging the direction of the axes.
Let the equation be
aa? +bxy + cy* + dx + ey +/=Q (1) ;
i* x = x'co&0 — y sin 5,
y = x f sin -f y' cos 0,
en (1) becomes
x*{a cos*0+csin*0 + &sin0cos0)
+ y' 8 (a sin* + c cos 8 5 — b sin 5 cos 0)
+ a?y{2(c-a) sin0cos0 + i (cos 8 0- sin 2 0)}
*'(<?cos0 + esin0)+y' (ecos0-<7sin0) +/=0 (2).
b
Now let tan 20 =
a — c
■en the coefficient of x'y' in (2) vanishes, and as in Art. 271
Bie coefficients of x 2 and y* are
4[a + c±V{(a-^) 2 + J 8 }].
fee of these coefficients must therefore vanish since their
■oduct is — - — > which, by hypothesis, =0; suppose the
Sufficient of x** = 0, thus, by suppressing accents on the
■Babies, (2) may be written
Cy* + I)x + Ey+f=0 (3).
If D be not = 0, this may be written
n f EV ' / E* )
cL thus the locus is a parabola. (Art 125.)
234 CEXTItE OF A CURVE OF THE SECOND DEGR3
If D = 0, then (3) represents two parallel straight li
one straight line, or an impossible locus, according a
greater, equal to, or less than 4Cf.
Hence if V — iac = the equation
ax 1 + bxy -f cy* + dx + ey +f=
represents a parabola subject to three exceptions, in w
rejjrciiL-nta respectively two parallel straight lines, one s
line, and an impossible locus.
By combining this result with those enumerated hi A
272, we have a complete account of the general equation 4
the second degree.
276. We have shewn in Art. 270, that when V
not = 0, the general equation of the second degree ref
a. central curve; we shall now prove that when b'—iac
the curve has not a centre except when the locus cwwwdj
t.ico parallel straight lines.
If a curve of the second degree hare theortg... „,
for its centre, no term involving tin first power of either^
variables alone can exist in the equation.
For if possible suppose that the origin of co-ordmits
the centre of the curve
ax* + bxy + cy* + dx+ey+f=0..
and let x t , y, be the co-ordinates of a point on the curve,
therefore — a;,, — y } co-ordinates of another point on the cum
substitute successively in (1), then
ax* + ox $i + c Vi + ^i + e l/, +f~ 0]
ax' + bx x y l + cy* — dx t — ey l +f=
therefore, by subtraction,
2 (dx 1 + cy l ) =0
Now unless d ande both vanish, (2) can only be truewW
{x 1 ,y^} lies on the line
dx + ey = 0.
But the centre of a curve is a point which bisects ««J
chord passing through it; hence the origin of
-
TKAC1NG A CUEVE OF THE SECOND DEGREE. 235
■mot be the centre of the curve (1) unless both d and e
arish.
277. Suppose then that we have an equation
ax*+bxy + cif + dx + ey+f=0 (1),
gwhich b* — 4ac = 0. Here a and c cannot both be zero, for
fen b would also be zero, and (1) would not be an equation
_.the second degree; we shall suppose that a is not zero.
WW if the curve denoted by (1) had a centre, and we took
Irt centre as the origin of co-ordinates, the terms involving
u first power of x and y would vanish bv Art. 276. But from
ate. 270 and 274 it follows that when & — 4oc = 0, we cannot
Exeral make these terms vanish by changing the origin or
ces. The only exception that can arise is when the nume-
in the values oi h and k in Art. 270 vanish, so that
* values of h and h become indeterminate, and the two
[nations for determining them reduce to one; see Algebra,
lapter xv. We have then 2ae — bd = 0, so that e = — .
Slice, by substituting for c and e, the equation (1) becomes
• i JV i bd *
ax* + bxy + -^ + dx + — y +/= 0,
*is, .(« + g , + rf(. + |*) + /-0 (2).
[nation (2) will furnish two values of x + tt- > s° that if
2a
sse values are possible the locus consists of two parallel
Bight lines. In this case any point on the line which is
tallel to these two and midway between them will be
centre.
Thus the result enunciated in the beginning of Art. 276
proved.
278. We may observe that relations similar to those
tained in Art. 274 hold when the axes of co-ordinates are
lique. For suppose the equation
ax* + bxy + cy* +/' =
236 TRACING A CURVE OP THE SECOND DEGREE.
to be referred to rectangular axes, and let the axes he t«
formed into an oblique system inclined at an angle © ; s
pose moreover that the new axis of x coincides with the
axis of x. We have then to put (Art. 84)
x = x' +y' cos a>, y = 3/ sin g> ;
substitute these values in the above equation and it becon
where a' = a,
V = 2a coa to + b sin g>,
c' = a cos 5 g> + S sin a> cos g> + c sin 2 o ;
thus b'* — 4a'c' = (J 2 — 4ac) sin 2 e»,
and a' +c' — V cos g> = (a + c) sin 2 © ;
so that — r-j = o — 4ac,
sin o>
, a! + c' — b' cos o>
and :— = = a + c.
sin a)
Therefore, by means of Art. 274, we conclude that fo
system of axes, rectangular or oblique, the expressions
b' 2 — Adc , a 1 ' + c — V cos o>
— s— a and — «
sin o> sin ©
remain unchanged when the axes are changed.
See Salmon's Conic Sections, 3rd Edition, pages 142
279. We shall now shew how to trace a curve c
second degree from its equation without transformati
co-ordinates; the axes may be supposed oblique or
angular.
Let the equation be
ax* + bxy + cy 2 + dx + ey +f=0 I
TRACING A CUEVE OP THE SECOND DEGEEE. 237
ilve the equation with respect to y ; thus
^±i{0*+«) , -*>("*+«fc+/)}*
2c ~ 2c
te±± ± L f(f-.4ac) a? + 2 (he-2cd) x + f- 4c/}*... (2)
X
+ fi±!f^v+v*+*)} (3),
L R = - —
2c ' P 2c '
5e — 2cd __ e* — 4cf
Suppose J 2 — 4ac negative, and write — /i for — jt — ;
[3) becomes
y = ax + /3±{-/i{a? + 2px + q)} i (4).
ow a? + 2px + q = (x+p)* + q—p*;
m q—p* be positive, the quantity under the radical is
ive and the locus impossible ;
if q — j?* = 0, the locus is the point determined by
x = -p, y = aa? + £;
if q — p* be negative, we may put
+p)* + q-p*={x+p + */{p*-q)}{x+p-*J(p 2 -q)}
= (x — 7) [x — 8) suppose ;
hus (4) may be written
y = ax + /3±{-fi{x-y)(x-8)} i (5).
iince (a?— 7) (a;— S) is positive, except when x lies between
1 8, the values of y in (5) are real only so long as x lies
een 7 and 8. Moreover y is always ^mte, and thus the
1 represented by (5) is limited in every direction.
238
TRACING AN ELLIPSE.
Since we know from our previous investigations that (j)
must represent one of the curves enumerated in Art 269, i
follows that it must represent an ellipse.
From the form of equation (5) we see that the chov
parallel to the axis of y are bisected by the line
y = ax + /3 (6).
For let there be two points on the curve (5) haying 1
common abscissa x l7 and the ordinates t/ 9 y", respectively; i
let y x be the corresponding ordinate of (6),
then y t = ouc x + £,
2 / = aa ?1 + y3 + {-^(a? 1 -7)(a? 1 -S)} i ,
^ = ^ + ^-{-^(^-7)^-8)}*.
Thus y^W + y");
and therefore the point (x l9 y x ) lies midway between the poi
(x t ,i/) andfo,y')-
I
TRACING AN HYPERBOLA.
In the figure DCD represents the line y= ax + 0; the
sisste of D' and D are 7 and S respectively; supposing
heater than 7. The centre G is midway between U and
its abscissa is therefore ^(7 + S). The equation to the
will give the ordmates of I)', D, G', G. Since GG' is
2I to the chords which D'D bisects, DD' and GG' are
gate diameters. GG' is a known quantity since the
iates of G and G' are known. DD is also a known
tity since the abscissas and ordinates of D and D' are
ra. The angle between GG' and DD' is known from
5 equation to DD'; the axes of the ellipse may therefore be
md (Arts. 193, 195).
II. Suppose b'~ iaa positive; put p for——,- ; thus
nation (3) becomes
3 = ax + P±yt(a? + 2px-i-q)}l (7).
Now x* + 2px + q = (x +p)' + j -p' ;
len q~p* be positive, the quantity under the radical is
^ays positive, whatever positive or negative value be as-
Bed to x. The curve therefore extends to infinity. Also
nay be shewn as before, that the line
1 diameter of the curve ; but it never meets the curve, be-
ise the quantity x' + 2px + q or {x+pY + q—p* cannot
lish. Hence the curve consists of two unconnected branches
ending to infinity, and is therefore an hyperbola.
If 2— p*=0, (7) becomes
y~ax + 0±^fL{x+p).
e locus now consists of two intersecting lines.
If q— p' be negative we may as before write (7) in the
m
y =ax + fi ± {fi {x-y) (x-S)}K
Hence x may have any value, positive or negative, except
we between 7 and 8; hence the curve consists of two
241
_t c is not zero, and as
lTO, it was not necessary
j. zero while considering
■ consistently with b*—4ac
nine the consequences of
; with respect to x instead
. be found on investigation
iv hen b' — iao is positive,
rid c are not both zero ; the
Suppose then a =
put in the form - Ns2LJ\
■ ■■ with the new axea*fl
— 0, and then it "becomes
\Yhen a=0 and c = 0, the
conclude that when b* — -lac
ays represents an hyperbola,
■od*-bed=0,
;cting atraight lines.
240 TRACING AN HYPERBOLA.
unconnected branches extending to infinity, and is thercta
hyperbola.
We shall be assisted in drawing an example o
by ascertaining the position of the asymptotes.
The equation to the curve i8
Expand by the Binomial Theorem ; thus
Z/^ + ,3 + W/*{l +£ (J + J) +&cj
The terms included in the &c, involve negative pow
x, and may therefore be made as small as we please lij
ficiently increasing x ■ hence from tlie nature of an asym
the. required equations to the asymptotes are
y = ax + fi + *Jti.{x+p),
and y = ax + f3~ \f/t(x+p).
Hence we can draw the asymptotes, and therefore tin.'
for they bisect the angles between the asymptotes,
intersection of the asymptotes is the centre, and tint
situation and form of the hyperbola are known.
The expression
, (e* - 4c/) (6' - Aac) - (he - 2cd)'
IV- (b'-iac)'
this vanishes when
(e' - icf) (b' - iac) - (be - 2cdj" = 0,
and therefore when
(b' - 4oc) /+ ae' + cd' - led = ;
so that if this relation holds the locus represented I
consists of two intersecting straight linea.
TRACING A PARABOLA. 241
We have hitherto supposed that c ia not zero, and as
4ac cannot be negative if o be zero, it' was not necessary
dvert to the possibility of c being zero while considering
first case. But as e may he zero consistently with b* — 4ac
ig positive, we most now examine the consequences of
oosing c zero.
The equation (1) may be solved with respect to x instead
rith respect to y. Hence it will be found on investigation
t the results hitherto obtained, when b 1 — iac is positive,
certainly tree provided that a and c are not both zero ; the
er case requires further examination. Suppose then a =
.c = 0; thus (1) becomes
bxy + dx+ey +f= ;
changing the origin this can be put in the form ,.^ i\A0 ,s \
bx'y'+f = 0, ~" ji \
-, bf-de \r. :; "'; y^/ £§
curve ia therefore an hyperbola with the new axesiftrlfs
mptotes, except when bf— de = 0, and then it becomes
> intersecting straight lines. When a=0 and c = 0, the
•ression
(J° — iac)f+ ae* + cd 3 — bed
aces to b (bf— de) ; thus we conclude that when 5° — iac
Joaitive the equation (1) always represents an hyperbola,
ept when
(6' — iac)f+ <xe a + cd 1 — bed — 0,
. then it represents two intersecting straight lines.
Til. Suppose b* — iac = 0, then (2) becomes
» — ^r * h I s < fe " 2ci) * + '' ~ le/1 ''
ich may be written
y = xc + {3±±(p'x + q')h,
T. C. S. IS
TRACING) A PARABOLA,
"~ 2c' " 2c'
p' = 2 {be - 2cd) , q=t?-±cf.
If p be positive, tlie expression under tlie radical ispofl-
tive or negative, according as # is algebraically greater or 1 -"
than — %■; if p' be negative, the statement must be reverai
In both cases the curve extends to infinity in one 3in9
only and is therefore a parabola.
The line y = ax + is a diameter, bisecting all ordinate
parallel to the axis of y, and meeting the parabola ,; I
poiut for which x = — —,.
If p = 0, the equation becomes
this equation represents two parallel straight lines it" ^' »
positive, and one straight line if q = ; if q' ia negative, ^
locus is impossible.
We have hitherto supposed in considering the third <*
that c is not zero; if e = 0, then i = 0, since £'-ta=*l
hence a and c cannot both be zero, for the equation (!) *
supposed to be of the second degree. As before, n m 1 ?
solve equation (l) with respect to x, and thus determine il*
peculiarities which occur when c = 0. We have tumuli'''
example when c is not zero, that the locus will consist *
two parallel straight lines, when
be — 2cuJ = 0, and e ! — 4c/is positive;
Lin like manner, if a be not zero, we can shew that the !«* I
will consist of two parallel straight lines when
bd— 2ae = 0, and <F — 4af is positive.
I
EQUATION OP THE SECOND DEGREE* 243
means of the relation b 2 — 4ac = 0, it is easily shewn that
second form of the conditions coincides with the first
q a and c are both different from zero. When a = the
is the necessary form of the conditions, but we see that
3econd form will then also hold. When c = the second
le necessary form, though the first will then also hold,
ce we shall include every case by stating that both forms
ie conditions must hold.
Similarly the conditions under which the locus will con-
of one straight line, or will be impossible, may be in-
igated.
J80. We will recapitulate the results of the present
)ter with respect to the locus of the equation
am? + bxy + cy 2 + dx + ey +/= 0.
L If b 2 — 4ac be negative, the locus is an ellipse admitting
ie following varieties :
(1) c = a, and — = cosine of the angle between the axes ;
s a circle (Art. 104).
(2) (e* — 4cf) (b 2 — 4ac) — (be — led) 2 positive ; locus im-
ible.
(3) (e* - 4c/) {b 2 - 4ac) - {be - 2cd) 2 = ; locus a point.
EL -If J 8 — 4oc be positive, the locus is an hyperbola,
apt when
(6 2 - 4ac) /+ ae 2 + ccP - bde = 0,
then it consists of two intersecting straight lines.
EH. If S* — 4ac = 0, the locus is a parabola, except when
•2cd = 0, and bd— 2ae = 0; and then it consists of two
illel straight lines, or of one straight line, or is impossible,
>rding as e 8 — ±cf and cP—kaf are positive, zero, or
ative.
16— %
244 EXAMPLES OF THE EQUATION OF TIIE SECOND I
EXAMPLES.
1. Find the centre of the curve
a:" — ±xy + A.y 2 — 2ax + iay = 0.
2. Find the centre of the ellipse
^('-S+K'-lH-
3. What i9 represented "by aa? + Ibxy + cy* =
4. Find the locus of the centre of a circle inscribed
sector of a given circle, one of the bounding radii of
sector remaining fixed.
5. In the side AB of a triangle ABC, any point
taken, and PQ is drawn perpendicular to AG; fini
locus of the point of intersection of the straight line;
and CP.
6. DE is any chord parallel to the major axis A
an ellipse whose centre is V; and AD and CE intersect
shew that the locus of P is an hyperbola, and fine
direction of its asymptotes.
7. Tangents to two concentric ellipses, the directii
whose axes coincide, are. drawn from a point P, am
chords of contact intersect in Q; if the point P alway
on a straight line, shew that the locus of Q will be a
angular hyperbola.
8. What form does the result in the preceding ex;
take when two of the axes whose directions are coincide:
equal?
9. Prove that an hyperbola may be described b;
intersection of two straight lines which move paral"
themselveswhlle the product of their distances from a
point remains constant.
»LES OF THE EQUATION OP THE SECOND DEGREE, 245
Two lines are drawn from the focus of an ellipse in-
; a constant angle; tangents ate drawn to the ellipse
points where the lines meet the ellipse ; find the locus
intersection of the tangents*
Find the latus rectum of the parabola
(y — x)* = ax.
Shew that the product of the semi-axes of the ellipse
y + 5x*=2 is 2.
Find the angle between the asymptotes of the hyper-
/ = bx* + c.
Find the equation to a parabola which touches the
- x at a distance a, and cuts the axis of y at distances
irorn the origin.
If two points be taken in each of two rectangular
30 as to satisfy the condition that a rectangular hyper-
Lay pass through all the four, shew that the position of
perbola is indeterminate, and that its centre describes
e which passes through the origin and bisects all the
rhich join the points two and two.
, Two lines of given lengths coincide with and move
two fixed axes in such a manner that a circle may
j be drawn through their extremities ; find the locus of
ntre of the circle, and shew that it is an equilateral
x)la.
, A variable ellipse always touches a given ellipse, and
common focus with it; find the locus of its other focus,
len the major axis is given, (2) when the minor axis is
. Draw the curve
jf-hxy + 6a?- 14a; + 5y + 4 = 0.
. Draw the curve
x* + y* — 3 (# + #)— xy = 0.
. Find the nature and position of the curve
y*- Sxy + 25a? + 6cy - 42ca? + 9c f = 0.
246 EXAMPLES OP THE EQUATION OF THE SECOND DEOEEL
21. The equation to a conic section being
ax' + 2bxy + cy* = 1,
shew that the equation to its axes ia
xy(a-c)=b(x*-y').
22. The locus of the vertices of all similar triangles wlios
bases are parallel chorda, of a parabola will in general \t
another parabola; but if any one of the triangles touch the
parabola with its sides, the locus becomes a straight line.
23. A series of circlea pass through a given point 0,
have their centres in a line OA, and meet another line BC.
Let Mho the point in which one of the circles meets the line
OA again, and let N be either of the points in whi
circle meets BC. From jl/and N lines are drawn parallel to
BO and OA respectively, internetting in P; shew that the
locus of -P is an hyperbola which becomes a parabola when
the two lines are at right angles.
24. The chord of contact of two tangents to a paraboh
subtends an angle fi at the vertex ; shew that the loens of
their point of intersection i3 an hyperbola whose asymptois
are inclined to the axis of the parabola at an angle <fr such
that
tan <f> = \ tan £.
25. Determine the locus of the middle points of tie
chords of the curve
ax 1 + ibxy + ctf + 2ex 4- %fy + g = 0,
which are parallel to the line x sin 6 ~ y cos 6 = 0; and hence
find the position of the principal axes of the curve.
26. Shew that the equation
represents two ellipses.
( 247 )
CHAPTER XIV.
MISCELLANEOUS PROK>SITIONS.
281. We shall give in this chapter some miscellaneous
•opositions for the most part applicable to all the conic
actions.
To find the equation to a conic section, the origin and axes
\ing unrestricted in position.
Let a, b be the co-ordinates of the focus; and let the
juation to the directrix be
Ax + By + (7=0.
"he distance of any point (x, y) from the focus is
ad the distance of the same point from the directrix is
Ax + By+C
Let e be the excentriaity of the conic section; then if (a?, y)
e a point on the curve, we have, by definition,
((*-«>•+ to-w t -^jf^> .... (Ih
,. (.-«.+{, _w-ite+$+«£ (2) .
We see from (1) that the distance of any point on a conic
action from the focus can be expressed in terms of the first
248 TANGENT TO A CURVE OF THE SECOND DEGItl
power of the co-ordinates of that point whatever he the oi
and axes. This is usually expressed by saying the dittax
of any point from the focus is a linear function of the <■"-■.■''■■ 1
nates of the point.
282. It will he seen by examining the equations to the
conic sections given in the preceding chapters that any conic
section may be represented by the equation
y* = mx 4- nx 2 .
The origin is a vertex of the curve and the axis of a an
axis of the curve; in is the latus rectum; in the paraboli
n=0; n is negative in the ellipse and positive in the hyper-
bola. In the circle m is the diameter of the circle and
n = ~\.
283. To find the equation to the tangent at any point of
a curve of the second degree.
Let the equation to the curve be
<tJ? + bxy + cy 1 + dx + ey +f= (t),
the axes being oblique or rectangular.
Let x, y be the co-ordinates of the point,
x", y" the co-ordinates of an adjacent point on the curve.
The equation to the secant through these points i3
y-y-
£(*-«0»
..(2).
Since (x', y') and (x", y") are on the curve,
ax* + bx'y + cy* + dx' + ey +/= 0,
ax" 1 + bx'y" ■+ cy"" + dx" + ey" +f= ;
.-. a (x"* - a") + b (x"y" - x'y') + c (y"* - y")
+ d(x"-x) + e(y"-y')
NOBMAL TO A CURVE OF THE SECOND DEGREE. 249
(*"- x') {a (*" + *') + hy" + d\
+ (f-y'){c(y"+y) + bx' + e} = o ;
. 9"~tf = a(x" + x') + bf + d
• % x"-x' c{y" + y') + bx' + e '
Hence (2) may be written
y — y=z i-Tj f SL- (x — X ).
* c(y" + y) + ia;' + e v '
Now in the limit x" = a?' and y" = y' ; hence the equation
he tangent at the point (#', y') is
'-'--^t&i*"- '> <3) -
This equation may be simplified ; we have by reduction
cy' + ta' + e) + x (2ax' + &y' + d)
= y'$cy' + bx'+ e) + x'{2ax' + by' + d)
= 2 (as* + bx'y' + cy" '+ dx' + ey' +/) - dx'-ey - 2/;
y (2cy' + &»' + e) + a; (2aa?' + 6y' + eZ) + da>' + ey' + 2/= 0,
(4).
Ify= 0, the curve passes through the origin, and the equa-
l to the tangent at that point becomes
d
y = --x,
ich we see does not involve the coefficients of a?, tf, or ay,
ke equation to the curve.
284. The. equation to the normal at the point (a/, y')
n the curve is expressed by equation (1) of the preceding
de and the axes are rectangular, will be
, 2cy ' + bx' + e f , N
y-y = 2ax' + br,' + d ( X - x) '
250 CHORDS SUBTENDING A RIGHT ANGLE.
285. It may be shewn as in Art. 183, that if fro
point (h, h) two tangents be drawn to the curve expresse
equation (1) of Art. 283, the equation to the chord of
tact is
y (2ck + bh + e) + x (2ah + bk + d) + dh + ek + 2/= 0.
286. All chords of a conic section which subtend a i
angle at a given point of the curve intersect in the norm
that point.
Take the given point of the curve as the origin of a
tern of rectangular axes, and let the equation to the curve
aa? + bxy -\- cy % -\- dx -\- ey = (1
The axis of x meets the curve at the points founc
making y = in the above equation, that is, at the p
x = 0, and x = .
7 a
Similarly the axis of y meets the curve at the origin
e
also at the point for which y = .
Hence the equation
a c
or ^+2- + 1 = (!
a e
represents the chord joining the points of intersection o:
axes and curve.
Also the equation to the normal to the curve at the o
is by Art. 284,
y=d x (<
Hence (2) and (3) meet in the point whose co-ordinate
— d —e
a + c' a + c*
• SEGMENTS OP A FOCAL CHORD. 251
3 whose distance from the origin is therefore
a + c
Now change the directions of the axes preserving the same
jfin ; the equation (1) will then become
dx" + b'x'y' + c'y' 2 + d V + e'y' = 0.
Also it appears from Arts. 274 and 275, that
a' + c' = a + c, and d* + e* = d? + e*.
Hence the normal at the origin will meet the new chord
the same distance from the origin as it met the original
rd, that is, will meet it in the same point Since this is
* whatever be the directions of the axes, it follows that all
chords intersect in the same point.
287. By comparing Arts. 154, 204, and 264, we see that
polar equation to anv conic section, the focus being the
a and the initial line the axis, is
I
r =
1+CCOS0'
ere Z= half the latus rectum.
We shall use this in proving the following proposition.
The semi-latus rectum of any conic section is an harmonic
tot between the segments made by the focus of any focal chord
Aat conic section.
Let A'SP= 0, see fig. to Art. 158 ;
1 -f e cos 6 *
Suppose PS produced to meet the curve again in P' ;
I
•\ SF =
1 + 6008(^ + 0)'
252 INTERSECTION OF TWO TANGENTS.
1 1 1+6COS0 1— 6COS0
,+ "77™ = 5 +
• • 8P l SB I ' I
"V
which proves the proposition.
288. The polar equation to the tangent to a conic
tion, the focus being the pole and the initial line the axi
(Art. 205)
-=ecos0 + cos (a — 0) (!
where a is the angular co-ordinate of the point of contad
Similarly the polar equation to the tangent at the
whose angular co-ordinate is ft, is
- = ecos0 + cos(/S — 6)
At the point where these tangents meet, we have
cos (a — 0) = cos (ft — 0).
Now we cannot have
since a and ft are by supposition different ; we therefore
a _ = _ ft y
Thus the two tangents (1) and (2) meet at the point'
angular co-ordinate is — — .
For example, suppose the conic section an ellipse;
ASP=a, ASQ = ft,
and let the tangents at P and Q meet at T\
TWO TANGENTS TO AN ELLIPSE. 253
• •
4b is, the two tangents drawn from any point to an ellipse
tiend equal angles at either focus*
Similarly the two tangents drawn from any point to a
imbola subtend equal angles at the focus.
With respect to the hyperbola we have to distinguish two
t^s. We have shewn in Art. 231, that from any point
Juded between the asymptotes and the curve, two tangents
fc be drawn both meeting the same branch of the curve, but
fcn any point included within the supplemental angles of
* asymptotes two tangents can be drawn meeting different
touches of the curve.
If now the two tangents from a point meet the same branch
an hyperbola, it may be shewn as in the case of the ellipse,
St they subtend equal angles at either focus; We will
tisider the' case in which the tangents meet different
anches.
254
TWO TANGENTS TO AN HYPERBOLA.
Let T be a point from which tangents TP 9 TQ are di
to different branches of an hyperbola.
Let A8P=a; and let the angle which QS prodi
through 8 makes with A 8 be £; then ft is an angle gre
than 7r, and ASQ = ft — it.
Thus the equations to TP and TQ will be respectivelj
- = e cos + cos (a — 9) , - = e cos + cos {JZ - 0).
At the point jT where they meet, we have
cos (a — 0) = cos (ft — 0).
We may therefore take = — — — , that is, we have -
as the angle which T8 produced makes with AS; thus
AST= it -
/. TSP^tt-^-,
TSQ=£^;
2
.\ TSP+TSQ = w;
POLE AND POLAK. 255
*t is, the angle which one tangent subtends at either focus
the supplement of the angle which the other tangent sub-
ids at the same focus.
289. We have given in Art. 120 the definitions of a pole
td polar with respect to a given circle. The same deuni-
ms are used generally substituting conic section for circle.
■ then the equation to the curve be
ax* ^bxy + cy 2 + dx + ey +f= 0,
ie equation to the polar of (x, y) is (Art. 283)
x(2ax' + by' + d) +y (2cy' + bx' + e) + dx' +ey' + 2/= 0.
290. If one straight line pass through the pole of another
Weight line, the second straight line will pass through the pole
tine first straight line.
Let (x\ y') be the pole of the first straight line, and
hffefore
p(2oa?' + %' + ^+y(2c/ + Jaj' + e) + ^ + ey + 2/=0...(l)
■ the equation to the first straight line.
Let (x" y y") be the pole of the second straight line, and
terefore
(2ax" + by" + d)+y (2cy" + bx' + e)
+ dx" + ey" + 2f=0 (2)
the equation to the second straight line.
Since (1) passes through (x", y") we have
QcT (2ax f +by' + d) +y" (2cy' + bx +e) + dx' + ey' + 2/=0,
at is,
f (2ax" + by" + d) + y' (2cy" + bx" + e) + dx" + ey" + 2/= ;
*ice (2) passes through (a?', y').
291. The intersection of two straight lines is the pole of
e line which joins th$ poles of those lines. See Art. 122.
256
QUADRILATERAL IN A CONIC SECTION.
292. If a quadrilateral ABCD be inscribed in a
section, of the three points E, F, Gr, each is the pole of th
joining the other two.
Let E be the origin ; EA, ED the directions of the
of x and y ; and let the equation to the conic section be
oaf + bxy + cy* + dx + ey +/=
Also suppose
EA = h, EB = h\
ED = h, EC=k'.
The equation to AG is t + % = 1
h ' k 1
.5Z)...J + |-1
■ AD -hh l
CJB , . . -p + j-, = 1
From (2) and (3) it follows that the equation
X *
(
represents some line passing through G. But from (4
(5) it follows that (6) represents some line passing throu
Hence (6) must be the equation to FG*
)NIC SECTION BEFERRED TO TANGENTS AS AXES. 257
ppose In (1) that y = ; then we have the quadratic
aa? + dx+f=0;
e roots of this equation are h and h' ; hence
a a
1 1 _ d
xnce (6) becomes
d# + ey + 2/=0,
it this, by Art. 289, is the equation to the polar of the
; therefore FG is the polar of & Similarly EG is the
of F. Hence, by Art. 291, G is the pole of EF.
'3. To determine the form of the general equation to a
section when the axes are tangents.
it aa? + lxy + cif + dx + ey + f=0 (1)
\ equation to the conic section.
> find where the curve meets the axis of x y put y =
above equation ; thus
aa? + dx+f=0.
the axis of a? is a tangent to the curve it must meet the
in only one point (see Art. 171) ; hence the roots of the
quadratic must be equal ; therefore
<P = 4a/. (2).
milarly that the axis of y may be a tangent to (1) we
have
e a = 4c/. (3).
C.S. 17
258 CONIC SECTION REFEliEED TO TANGENTS AS AXES.
Substitute the values of a and c from (2) and (3), then (
becomes
dV + idfx + e y + Asfy + ibfry + 4/V 0,
or (dx + ey + 2/)' + (4bf- 2de) xy = 0,
d e A' W-de
Iut V ,= "4' V"~»* ^""^
thus we ohtain for the required equation
u
■f-^
pxy .
By puttin? successively x and y = 0, we see that A is I
distance from the origin to the point where the curve, me
the axis of x, and k is 'the distance from the origin to 1
point where the curve meets the axis of y.
If it be required to determine a conic section which toad
two given straight lines in given points, and ako p«s
through another given point, we may assume the last writ
equation to represent it, so that the lines to be touched:
taken as the axes of x and y ; then by putting the co-or
nates of the additional given point in the equation we fi»
single value for pi. Thus there is only one conic
satisfying the data.
me Mt
294, Suppose the equation
(M- 1 )"+^=»
to represent a parabola. Then, by Art. 280,
PARABOLA REFERRED TO TANGENTS AS AXES. 259
If /a=0, (1) becomes
is equation represents the straight line joining the points of
•ntact of (1) with the axes.
If /* = — tt , we have from (1),
(H-0-3 »•
X _
V(D + f= ij
We may write this
vWi =1 (8) '
membering that the radicals may be positive or negative.
hns (3) is the equation to a parabola referred to two tan-
*nts as axes.
295. We may notice the form of the equation to the
ngent to the parabola
s/hjh* «•
The equation to the secant through (as', y') and (x", y") is
Since (a;', y') and (x", y") are on the parabola, we have
17—2
8IMIIAK CUKYE3.
Hence the equation to the secant may be written
Hence we have for the equation to the tangent at
Vl« « VM i/M
Similar Curves.
tt^.yi
_
' 296. Def. Two curves are said to be similar and «""
farfy situated when a radiu8 vector drawn from some as
point in any direction to the first curve bears a constant rat
to the radius vector drawn from some fixed point in a parall
direction to the second curve.
Two curves are satd to be similar when a radius veel
drawn from some fixed point in any direction to the first cm
hears a constant ratio to the radius vector drawn from aoi
fixed point to the second curve in a direction inclined al
constant angle to the former.
The two fixed points arc called centres of similarity.
297. If two curves are similar, so that a pair of centres
similarity exists, then an infinite number of pairs of centres
similarity can be found.
ALL PARABOLAS ABE SIMILAR. 261
For, suppose 0, O to denote one pair of centres of stmt"
ity ; ana let OP, OQ be radii vectores of the first curve,
1 OP, OQ the corresponding radii vectores of the second
:ve, so that the angle PO Q = the angle P O Q, and
OP _ OQ
OF~ OQ*
Suppose any point S taken and joined to ; then make
) angle P OS' = the angle POS, the angles being measured
the same direction, and take OS' so that
OS' _ OP
08" OP 9
m 8 and 8' shall be centres of similarity.
For join SP, SQ, S'F, S'Q; then the triangles SOP,
OP are similar; and so also are the triangles 80 Q,
OQ. Hence it easily follows that
the angle QSP=QS'P;
A SP - a Q
WF~B'Q'
d thus the proposition is established.
298. All parabolas are similar curves.
Let 4a be the latus rectum of a parabola, and 4a' the latus
stum of a second parabola. The polar equations of these
rves, the foci being the respective poles, are
_ 2a
, 2a
~" 1 + cos ff *
Hence, if = ff, we have
r a
-? = ->•
r a
ros any two parabolas are similar, and the foci are centres
similarity.
■
ively
lined, h
264 CONDITIONS OF SIMILARITY.
300. Next, suppose we require the curves (I) and (i)
Art. 299 to be similar without the limitation of being li
larly situated. For x and y in (1) we put respectively
Tt + r cos 8, k-\-r sin 8,
For x and y in (2) we put respectively
K + r' cos {8 + a), k' + r' sin (0 + a),
where a is some constant angle at present undetermined. I
ceed as in Article 299; instead of equation (7) we shall no
have
a co3*g + b sin 8 cos 8 + c sin' 6
a cos" [8 + a) + b' sin {8 + a) cos [8 + a) + c' sin' (8 + a)
= a constant = ^ a
This may be written
I a cos'0 + Z> sin 8 cosB + c sin*0 _
^ cos'0 ~+B sin cos + sin* 8 ~ /i '
where
A = a cos'a + c' sin'a + b' sin a cos a,
5=2 (c'— a') sin a cosa + b' (cos'a — sin'a),
(7= a sin'a + c' cos' a — b' sin a cos a.
I
That the curves may be similar we must have
ABO
g (A + Of
■■V (a+cf ;
g f
■• (^+0)'- (.+.)-
ac (a + c)'
mce,
MISCELLANEOUS EXAMPLES. .265
AG _ ac
&-1AC b*-<4a c
{A+C)* ~(a + c) 2#
But A + (7= a' + c',
I P 1 - 4^4 (7 = V* - 4aV, (Art. 274) ;
&"-4aV _ y-4ac
-# * (a' + c') 8 ""(a + c) 2-
This relation must therefore hold, in order that the given
ves may be similar.
EXAMPLES.
1. Straight lines are drawn through a fixed point ; shew
t the locus of the middle points of the portions of them
arcepted between two fixed straight lines is an hyperbola
ose asymptotes are parallel to those fixed lines.
2. Through any point P of an ellipse QPQ is drawn
rallel to the axis major, and PQ and PQ each made equal
the focal distance SP, find the loci of Q and Q*.
3. In the given right lines AP, A Q, are taken variable
nits p, q, such that Ap : pP :: Qq : qA ; prove that the
m of the point of intersection of Pq and Qp is an ellipse
rich touches the given right lines in the points P, Q.
4. TPj TQ are two tangents to a parabola, P, Q being
> points of contact ; a third tangent cuts these in p, q
pectively ; shew that
TP* TQ #
5. TP, TQ are equal tangents to a parabola, P, Q being
points of contact ; if PT, QT be both cut by & third
gent, prove that their alternate segments will be equal.
266. MISCELLANEOUS EXAMPLES.
6. From a point are drawn two lines to touch i
bola in the points P and Q ; another line touches the pi
in R and intersects OP, OQ, in S and T; if Fbe the:
section of the lines joining Pl\ QS, crosswise, 0, E, Fare!
the same straight line.
7. From an external point two tangents are drawn to an
ellipse; shew that an ellipse similar and similarly situate!
will pass through the external point, the points of contact, in!
the centre of the given ellipse.
8. A and Sare two similar, similarly situated, and
centric ellipses ; C is a third ellipse similar to ,4 and B, ia
centre being on the circumference of B, and its tsxB pUW
to those of A or B; shew that the chord of intersection ot'i
and is parallel to the tangent to B at the centre of C.
9. The line joining any point with the intersection oft
polar of that point with a directrix subtends a right angle
the corresponding focus.
10. If normals be drawn to an ellipse from a given pool
the points where they cut the curve will lie on a rectangulu
hyperbola which passes through the given point and !i.- -■
asymptotes parallel to the axes of the ellipi
11. If CM, MP are the abscissa and ordinate of MJ
point P, on the circumference of a circle, and MQ is takei
equal to MP and inclined to it at a constant angle, the toe*
of the point Q is an ellipse.
12. Having given the equation to a conic section
aa? + ibxy + if +/= 0,
rind the locus of the intersection of normals drawn at tin
extremities of each pair of ordinatcs to the same abscissa.
13. Any two points P, Q are taken in two fixed lines is
one plane such that the line PQ is always parallel to a gi^n
line; P, Q arc severally joined with two n'xed points 2,^',
find the locus of the intersection of Pff and QR.
MISCELLANEOUS EXAMPLES. 267
. The tangent at any point Pof a circle meets the tan-
t a fixed point A in T 7 and T is joined with B the
rity of the diameter passing through A ; shew that the
>f the point of intersection of AP and BT, is an ellipse.
. The polar equation to a conic section from the focus
— c cos = J,
r
hat the equation to a straight line which cuts it at the
for which = a and £ respectively, is
1 A z fn OL+P\ OL-0
— c cos 5 = J cos 0-^ ^ sec — r-*-.
(-'-¥)
Chords are drawn in a conic section so as to subtend
;ant angle at the focus ; prove that the locus of the foot
perpendicular dropped from the focus upon the chord
xcle, except in a particular case when it becomes a
it line.
If SP, 8Q be focal distances of a conic section in-
j a constant angle ; shew that PQ touches a confocal
Having given two fixed points through which a conic
is to pass, and the directrix, find the locus of the
onding focus.
The focus and directrix of an ellipse are given ;
h the former a line is drawn making with the latter an
rhose sine is the excentricity of the ellipse. Find the
f the points where this line meets the curve, the excen-
being variable.
A series of conic sections is described having a com-
cus and directrix, and in each curve a point is taken
distance from the focus varies inversely as the latus
; find the locus of these points.
268 MISCELLANEOUS EXAMPLES.
21. Two conic sections have a common focus Sthroi| g
which any radius vector is drawn meeting the curves in P,
respectively. Prove that the locus of the point of intersect a
of the tangents at P, Q, is a straight line.
Shew that this straight line passes through the intersect
of the directrices of the conic sections, and that the sines
the angles which it makes with these lines are inversely pit
portional to the corresponding excentricities.
22. A line is drawn cutting an ellipse in the points P,f
let Q be either of the points in which the same lino ma
similar, similarly situated, and concentric ellipse; shew ill
if the line moves parallel to itself, PQ . Qp 13 constant.
23. In two straight lines OX, OT, which intersect in ( "
take OA — a, OB — b ; shew that the centres of all the M '
sections which touch the lines in A and B lie on the strsi
line
ay m bx.
24. About two equal ellipses whose centres coincide,;
whose major axes are inclined to each other at a given angle
an ellipse is circumscribed ; if A and B be the semi-ass J
the circumscribing ellipse, a and b the semi-axes of the qn
ellipses, and 2a the inclination of their major axes, then mil
aV + A'B 1 = (A*b* + B'a') cos'a + (AV + BV) sin'a.
Hence shew that about the two equal ellipses a tin
ellipse may be circumscribed.
25. Two similar ellipses have a common centre and tone
each other; if n be the ratio of their linear magnitudes,"
the ratio of the major to the minor axis in either, and a ' "
inclination of their major axes, prove that
1
MISCELLANEOUS EXAMPLES. 269
26. Two tangents (a, b) to a parabola intersect in Pat an
le cd, and a circle is described between these tangents and
curve ; shew that the distance of its centre from P is
ab
(a + b) sec - + 2 tj(ab) tan -
27. If two chords at right angles be drawn through a
id point to meet a curve of the second degree, shew that
1 + 1
Rr ' RV
constant, where R and r are the segments of one chord made
the fixed point, and R' and / those of the other. ' M
28. The equation to the locus of the foci of all parabolas
tose chords of contact with axes inclined at an angle a cut
a triangle of constant area is
r = h ^{sin sin (a — 0)}.
29. A parabola slides between two rectangular axes, find
t curve traced out by the focus.
30. A parabola slides between two rectangular axes, find
B curve traced out by the vertex.
31. Successive circles are drawn each touching the pre-
ling one externally and each having double contact with a
ren parabola; shew that their radii form an arithmetical
agression whose common difference is the latus rectum.
32. A system of ellipses is represented by the equation
rectangular co-ordinates
aa? + Zcxy + by* = n (a + b) 9
bere a, i, c are variable and n constant ; shew that every
•rallelogram constructed on a pair of perpendicular diameters
diagonals will circumscribe a certain fi^ed circle.
270 MISCELLANEOUS EXAMPLES.
33. If from any point in the tangent to a conic at
perpendicular be dropped upon the line joining the focus ti
the point of contact, prove that the distance of the point ■
the tangent from the directrix is to the distance of the foot!
the perpendicular from the focus as 1 : e.
34. Upon a given straight line as latus rectum, let
number of conic sections be drawn, and from the focus
two straight lines be drawn intersecting them all; tlieatl
chords of all the intercepted arcs will, if produced,
through a single point.
35. A line of constant length moves so that its w
always lie on two given Sines; find the locus traced out 1
a point in the line which divides it hi a given ratio.
36. In any conic section if r and r be focal distances
right angles to each other, and I be half the latus rectum, :]«
(H+(H)*
37. Two conic sections equal in every respect are pi
with their axes at right angles and with a common foeffl ft
SP, SQ being radii vectores of the one and the other atn$
angles to each other, find the locus of the intersection of fit
tangents at P and Q.
Also find the locus when SPQ is a straight line.
38. S and iZare the foci of an ellipse, and round S, B,'
focus and centre, another ellipse is described, having its raiw
axis equal to the latus rectum of the former. Through jot
point P in the first draw SPQ to meet the second ; it is in-
quired to find the locus of the intersection of HP and tt<
ordinate QM.
39. A and B are the centres of two equal circles; Jfl
BQ, radii of these circles at right angles. If AB , = ii?>
the line PQ always passes through one of the points of inter-
section of the circles.
MISCELLANEOUS EXAMPLES. 271
Tangents are drawn to a conic section at the points P,
other tangent is drawn at an intermediate point Q, and
the other tangents in M, N; shew that the angle MSN
the angle PoR x S being a focus.
In a parabola the angle between any two tangents is
e angle subtended at the focus by the chord of contact.
A triangle is formed by the intersections of three
ts to a parabola ; shew that the circle which circum-
this triangle passes through the focus.
Given a focus and two tangents to a conic section,
hat the chord of contact passes through a fixed point.
A circle is described upon the minor axis of an ellipse
neter ; find the locus of the pole with respect to the
of a tangent to the circle.
In a parabola two focal chords PSp, QSq, are drawn;
hat a focal chord parallel to PQ will meetly produced
tangent at the vertex.
If from the vertex of a parabola a pair of chords be
at right angles to each other, and on them a rectangle
lpleted, prove that the locus of the further angle is an-
parabola.
, From a point P in the circumference of an ellipse
PQ, PR are drawn at right angles ; express the co-
tes of the point of intersection of QR with the normal
i terms of the co-ordinates of P. Shew that as P moves
the ellipse this point of intersection will describe the
a 2 " b
- ( a ~ b V
, Shew that the locus of the centre of an equilateral
x>la described about a given equilateral triangle is the
nscribed in the triangle.
<
272 MISCELLANEOUS EXAMPLES.
49. Two equal parabolas have the same axis
but are turned in opposite directions ; chords of one parabola
are tangents to the other ; shew that the locus of the middk - '
points of the chords is a parabola whose latus rectum is M
third of that of the given parabola.
50. The co-ordinates of the focus of the parabola whos
equation when referred to two tangents inclined at an an^tt
v©v®-
1, are
- , and -
51. Iiiaa?+2hxy + cy*+2ax + 2c'y + d = <) be the « t
tion to a parabola, the axis of the parabola will be given lj
the equation
<" +s >(* + ^) + < 4+c >(j' + ;^) =
52. Two equal parabolas have the same focus and thai
axes are at right angles to each other, and a normal to oneol
them is perpendicular to a normal to the other ; prove thai
the locus of the intersection of such normals is a parabola.
53. Find the locus of the intersection of two normals in
an ellipse which are at right angles.
54. Normals are drawn at the extremities of the conju-
gate diameters of an ellipse, and by their intersections fori
a parallelogram. If <£ denote the excentric angle of an ex-
tremity of one of the conjugate diameters, shew that the are*
of the parallelogram is
— — 7 — - sin tp cos' <p.
55. Through the four angular points of a given square i
circle ia drawn, and also a series of curves of the secW
order, and common tangents to the circle and each curve Hi
drawn. Find the locus of the points of contact of each cui«
with its tangent.
MISCELLANEOUS EXAMPLES. 273
56. From any point T outside an ellipse two tangents TP
d TQ are drawn to the ellipse ; shew that a circle can be
scribed with T as centre so as to touch SP, HP, SQ, HQ,
these lines produced.
If x and y are the co-ordinates of T t shew that the radius
the circle is
a
T. C. S. 18
{ 274
CHAPTER XV.
AMIIDliED NOTATION.
301. Through five points, no three of which are in
straight Hue, one conic wet-tun and onlg one ran be drawn.
Let the axis of x pass through two of the five points, aai
the axis of y through two of the remaining three points. Li
the distances of the first two points from the origin be h v I,
respectively, and those of the second two points l\, /.'.„ r.-
spectively ; also let h, k be the co-ordinates of the renuutn
point. Suppose (Art. 269)
ax* + bxy + cy" + dx + ey + 1 = (1)
to be the equation to a conic section passing through the fin
points. Since the curve passes through the points [li„
(A f , 0), we have from (1)
ak 1 1 +dh l +l=0 (2),
ah* + dk t + l = (3).
Similarly, since the curve passes through (0, k
we have
(*ty
ck*+ek 1 +l = (4),
ck,* +ek t +l = Q (5).
Lastly, since the curve passes through (h, k), we have
ah' + bhk + c&? + dh + ek+l = (6).
From (2) and (3) we find
«- 1 A h > + h >
•
CONIC SECTIONS THROUGH FIVE POINTS. 275
from (4) and (5) we find
_ 1 _ _ k, + k %
from (6) we can determine the value of i. Since no
e of the five given points are in the same straight line,
5 of the quantities h^ h 2 , k v & a , h> k, can be zero ; hence
values of the coefficients a, i, c, d, e are all finite. If
substitute these values in (1), we obtain the equation to a
c section passing through the five given points. As each
le quantities a, J, c, d, e, has only one value, only one
b section can be made to pass through the five given
ts.
K)2. The investigation of the preceding article may still
pplied when three of the given points are in one straight
; the point (A, k) for instance may be supposed to lie on
line joining (0, kj and (h v 0) ; the conic section in this
cannot be an ellipse, parabola, or hyperbola, since these
es cannot be cut by a straight line in more than two
ts ; the conic section must therefore reduce to two straight
i, namely the line joining the three points already spe-
1, and the line joining the other two points. If, however,
of the given points are in one straight line, the method
le preceding article is inapplicable; it is obvious that
\ than one pair of straight lines can then be made to pass
lgh the five points.
103. We shall now give some useful forms of the equa-
to conic sections passing through the angular points of a
gle or touching its sides.
jet u = 0, v = 0, to = be the equations to three straight
which meet and form a triangle ; the equation
Ivw + mwu + nuv = (1),
e Z, m, n are constants, will represent a conic section
ibed round the triangle ; also by giving suitable values
m, n 9 the above equation may be made to represent any
section described round the triangle. This we proceed
ove.
18—*
;,
27G CONIC SECTION PASSING THROUGH
I. The equation (l) is of the second degree in the varUW <1
x and y, which occur in the expressions u, v, w ; henca (
must represent a conic section.
II. The equation (1) is satisfied by the values of a
y, which make simultaneously i* = 0, w = 0\ the conic i
therefore passes through the intersection of the lines
sented by v = and w = 0. Similarly the conic section
through the intersection of w — and u = 0, and also t'
the intersection of « = and v = 0. Hence the conic
represented by (1) is described round the triangle
by the intersection of the lines represented by w =
w = 0.
III. By giving suitable values to /, m, n, the emuti
(1) will represent any conic section described round Inel
angle. For let S denote a given conic section described ra
the triangle ; take two points on S; suppose A„ k the «mj e
nates of one of these points, and A a , k t those of the other,
we first substitute A, and k, for a; and y respectively in (l)i '
then substitute h % and Jc a , we have two equations from wk
we can find the values of -y and -j ; suppose *y = p and j=(
Substitute these values in (1), which becomes
vw+pwu + quv = (S
this is therefore the equation to a conic section which k
Jive points in common with S, namely, the three migM
points of the triangle and the points (/j 1( A,}, (A,, I;,). H"
conic section (2) must therefore coiucide with S by Art.W .
Hence the assertion is proved.
We might replace one of the constants in (1) by
but we retain the more symmetrical form; (l) m»y
written
304. Equation (1) of the preceding article may be wriM
w (lv + mu)+nuv=*0 $ I
THE ANGULAR POINTS OP A TBI ANGLE. 277
will now determine where (1) meets the straight line
resented by
lv + mu-0 (2).
By combining (2) with (1) we deduce nuv = ; therefore
ler t* = 0, or v = 0; but by taking either of these suppo-
ons and making use of (2), we see that the other sup po-
em must also hold; hence the line (2) meets the curve (1)
only one point, namely, the point of intersection of u =
lt> = 0.
Hence (2) is the tangent to (1) at this point. Similarly
+ nv = is the tangent to (1) at the point of intersection
o = and v = 0, and nu + Iw = is the tangent at the point
ntersection of u = and w = 0.
305. The demonstration of the preceding article is imper-
t, because we know from Arts. 132, 222, that a line parallel
the axis of a parabola or to either asymptote of an hyper-
a meets the curve in only one point, but is not the tangent
that point. The proposition may however be established
the following manner. Take the axis of x coincident with
\ line u — 0, so that u becomes qy, where q is some con-
nt ; also take the axis of y coincident with the line v = 0,
that v becomes px, where p is some constant. Suppose
= Ax + By+C. Then (1) of the preceding article be-
nes
(Ax + By + C) (Ipx + mqy) + npqxy = 0.
By Art. 283 the equation to the tangent at the origin, that
at the intersection of x = and y = 0, is Ipx + mqy = 0, or
f mu = ; which was to be proved.
306. Let each of the three tangents in Art. 304 be pro-
ced to meet the opposite side of the triangle formed by the
es w = 0, t? = 0, tfl = 0; then it may be shewn that the
•ee points of intersection lie on the straight line
7 + — + -=0.
I m n
The lines joining the angular points of the triangle formed
the tangents with the angular points of the original
2(S CONIC SECTION PASSING THROUGH
triangle respectively opposite to them, are represented >;■' i'
equations
?-s-°' i-l-O' -n--r°-
these three lines meet in a point. Thus when a triangle
inscribed in a conic section the lines joining each point wil
the pole of the opposite side meet in a point.
307. Let w = 0, v = 0, w = be the equations to dun
straight lines, then the equation
Au* + Bv t + Cv? + lA'vw + IB'wu + 2 C 'uv =
will generally represent any assigned conic section, if tk
constants A, B, 0, A', B\ C are properly determined.
For suppose we divide the equation by one of the conntanl
as C, there are then five independent constants left. N»
let S denote any assigned conic sectiou ; take five points in J
and substitute the co-ordinates of the five points succesairclt
in the above equation; we shall thus have five equations f«
determining the five constants. Suppose a, b, c, a, V tie -.
value9 thus determined, then the equation
om* + h? + cu? + 2a'«w + 2b'tou + 2uv =
represents a conic section which has five points in common
with S, and which therefore coincides with S. (Art. 301.)
308, The method of the preceding article, although im-
portant and instructive, is not satisfactory, because we b»«
not proved that the five equations from which the constsnB
are to be determined are constsie7it and independent. There may
be exceptions to the theorem, and we therefore uae the worn
generally in the enunciation. If the three straight lines mw*
in a point, then the curve denoted by the equation always
passes through that point, and the equation in this case will
not represent any assigned conic section. If the three straight
lines are parallel, u, v, w take the forms
Ix + my + p, kc + my + p', Lc + my + p'\
and the equation takes the form
X (Ix + my) 1 + u- (Iv + my) + v
THE ANGULAR POINTS. OF A TRIANGLE. 279
rich represents two parallel straight lines, and thus will not
present any assigned conic section. With these exceptions,
sreTer, the theorem is universally true, as we shall now
ew by another demonstration.
Since the lines are not all parallel, two of them at least will
set ; suppose u = and v = to be these two, and take their
rections for the axes of y and x respectively ; then u =
comes x = 0, and v = becomes y = ; also w = may be
itten Ix + my + n = 0. We have then to shew that the
nation
&+By* + C(lx + my + w) 1 + 2A'y (lx + my + n)
+ 2B'x(tx + my+n) + 2C'an/ = (1)
11 represent any assigned conic section by properly deter-
ining the constants A, B, &c. Suppose
aa? + 2 bxy + cf + 2dx + 2ey +/= (2)
be the equation to the assigned conic section. Arrange the
tms in (1) and equate the coefficients of the corresponding
rms in (1) and (2) ; thus
Cn*=f, A'n + Cmn = e, B'n + Cln = d,
B+Cm* + 2A'm = c, A+CP+2B'l = a,
Clm + A'l + B'm+C' = b.
These equations determine successively (7, A\ B, B } A, C\
9 the given lines do not meet in a point, n is not zero; hence
e values found for (7, A\ &c. are all finite and determinate.
hufl (1) is shewn to coincide with (2), and the required
eorem is proved.
309. To express the equation to a conic section which
uches the sides of a triangle.
Let t* = 0, v = 0, w = be the equations to the sides of a
iangle ; then any conic section may be represented by the
^nation
Au* + Bv 9 +Cu?.+2A'vv> + 2B'tmt + ZC'uv=*0 (1).
280 CONIC SECTION TOUCHING
To find where this conic section meets the line a = 0, we miai
putw = 0; thus (1) becomes
Bv > +Cw'+2A'vio = Q (2).
Now from (2) we obtain by solution two values of-, i
— = /*,) and — = /*,. The equation v=/i 1 w represents soi
straight line passing through the intersection of v =0, aoi
w = 0. Hence since (1) is satisfied by those values of lanl
y which make simultaneously u — and v—^w = 0, the inter-
section of the lines w = and v — /a,w = is a point on %
Similarly the intersection of u — and v — ft s w = is a paH
on (1). Hence the line w = will meet (1) in two pointful ^
therefore will not be a tangent to it, unless the lines
v — fijte = 0, and v — /i s w = 0,
coincide. Hence that u = may touch (1) we must W
/i 1 = /i a , and therefore A' i = BC.
Similarly that v = may touch (l) we must have B"= CA] I
and that w = may touch (1) we must have C" = j4B. Fran I
these three relations we sec that A, B, and G must have ik I
same sign, because the product of each two i3 positive. Also I
the sign of A, B, and C may be supposed positive,
if each of them were negative we could change tin
every term in (l), and thus make the coefficients of u\i J , j
and io a positive. We may therefore put
A^r, B = m\ C=n';
thus
A'=± mn, B' = ± nl, C' = ± In.
Hence (1) becomes
JV + mV+aV± '2mnvw± 2nlwu ± 2hmtv =
We shall now examine the ambiguity of signs that appears
in this expression.
I. Suppose all the upper signs to be taken. The eqaa-
tion may then be written
{lu + mv + nu>y=0.
THE SIDES OF A TRIANGLE. 281
This is the equation to a straight line, or rather to two
>incident straight lines.
H. Suppose the lower sign to be taken twice and the
piper sign once ; we have then three cases,
(lu + mv — nw)* — 0, or (lu — mv + nw) 2 = 0,
or (— lu + mv + nw) 2 = 0.
«ach equation represents two coincident straight lines.
III. Since then the forms in I. and II. represent straight
Jies, we see by excluding these cases from (3), that if a curve
t the second degree touch the straight lines
u = 0, v = 0, w = 0,
8 equation must take one of the forms
Pu* + m*v* + n*w* — 2mnvw — 2nlwu — 2lmuv = ... (4),
ZV + «iV + nW — 2mnvw + 2nlwu + 2lmuv = ... (5),
ZV + wV + nV + 2mnvw — 2n?wn* + 2lmuv = ... (6),
Pw f + iwV + nW + 2?wwvw? + 2nlvm — 2?wmv = ... (7).
These four forms may also be written
VW+ VM + *J(nw) =0 (8) from (4),
V(-&)+ V(«">) + V(»w) =0 (9) ... (5),
VW+V(-««)+ hj{nw) =0 (10) ... (6),
V(&)+ VM +V(-w^) = o (n) ... (7),
nrhich may be verified by transposing and squaring, so as to
rat the equations in a rational form.
310. It is easy to verify the proposition that the curve
•epresented by the equation
*J{lu) + *f(mv) + */(nw) =
annot cut the lines u = 0, v = 6, w = 0. For suppose the
282 CONIC SECTION TOUCHING THE SIDES OF X TRIANGLE.
above equation satisfied by the co-ordinates of a point ; then
these co-ordinates must make lu, mv y and nw, all positive, or
all negative. Suppose lu is positive ; then for any point on
the other side or u = 0, the expression lu becomes negative,
and thus the co-ordinates of such a point will not.Badtmjfa
equation unless both mv and nw are also negative. But if tk
curve cuts the line w= 0, there will be poipts on both sides of
u = lying on the curve, and it will be possible to change
the sign of w without changing the signs oft; and w. Hence
the curve cannot cut the line u = 0. Similarly it cannot cot
the lines v = 0, w = 0.
The same mode of proof will shew that the curves repre-
sented by equations (9), (10), and (11), of the preceding article
cannot cut the lines u = 0, v = 0, w = 0.
311. The forms in equations (5), (6), and (7) of ArtSto
may be derived from (4) by changing the sign of one of Ae
constants. Thus, for example, (5) may be derived from (4)
by changing the sign of I. In the following article we shell
use (4) as the equation to a conic section touching the sidee
of a trianele; it will be found that we might have used (5), (6).
or (7), We shall see in a subsequent article, a case in whki
it is necessary to distinguish the forms. See Arts. 314, 315.
312. Equation (4) of Art. 309 may be written
(lu — mv) % + nw (nw — 2mv — 2lu) = (1).
If we combine this with w = 0, we deduce that
lu-mv=*Q.. r (2);
hence we can interpret the last equation ; it represents a line
passing through the intersection of t*= and t> = 0, and alee
through the point where the line w = meets the curve (1)«
It may be shewn as in Art. 304, that
nw — 2mv — 2?w = -. (3)
represents the tangent to (1) at the other point where (2)
meets it.
CIRCUMSCRIBED CIRCLE. 283
Similarly we can interpret
mv — ni0 = O (4),
lu — 2nw — 2mv = (5),
nw — Zu = (6),
mv—2lu—2nw = (7).
*The intersection of (3) with w = 0, of (5) with & = 0, and
* (7) with v =s ^rUl lie on the line
lu + mv + nw s= o.
The line ?w + ?wv = passes through the intersection of
=*0 and v = 0, and also through the intersection of (3)
id w = ; hence its position is known.
Similarly mv + nw = 0, and nw + ?u = 0, can be inter-
reted.
313. 2b jftw? *Ae equation to the circle described round a
(angle.
It will be convenient in this and the two following articles
use the form
x cos a + ysina — p =
the type of the equation to a straight line ; we shall there-
re put a, & 7 for u, v> w respectively (Art. 73).
Let a = 0, ft = 0, 7 = be the equations to the sides of
triangle ; then, by Art. 303,
Iffy + my<i + na/3 = •••(!)
11 represent any conic section described round the triangle ;
mce by giving proper values to I, m, n, this equation may
; made to represent the circle which we know by geometry
n be described round the triangle. We might proceed
us: in (1) write for a, /J, 7 the expressions which they
present, then equate the coefficient of xy to zero, and the
efficient of a? to that of y s ; we shall thus have two equa-
>ns for determining j and -, ; and with the values thus
284 1NSCEIBED CIRCLE.
obtained (1) will represent the required circle. We Iot
this as an exercise for the student, and adopt another mclbaL
The equation to the tangent to (1) at the intersection of
a = 0, and ft = 0, is, by Art. 304,
lft+ma=0 (2).
Let A, B, C denote the angles of the triangle opposite tk
sides a = 0, ft = 0, 7 = 0, respectively; by Euclid, III. 3!,
the tangent denoted by (2) must make an angle A with lie
line a = 0, and an angle B with the line ft = 0. Suppose lit
origin of co-ordinates within the triangle, then the equation
to the line passing through the intersection of a = wid
ft = 0, and making angles A and B respectively with thi*
lines, is
«sm5 + /3sm4 = (3).
Thus (2) must coincide with (3) ; therefore
I sin A
Similarly, ™ = sirTC'
a the trr I
Thus the equation to the circle described round
angle is
fty sin A + ya sin B + aft sin C = 0.
314. To find the equation to the circle inscribed in «
triangle.
Suppose the origin of co-ordinates within the triangle ; thai
for all points on the circle a, ft, y are negative quantiti*
(see Art. 54). Now the equation to the circle must be of ow
of the forms (8), (9), (10), (11) given in Art. 309 ; the first*
the only form applicable, namely,
V(fc)+VH9) + V(«7)=o (1),
which is equivalent to
V(-i«) + V(— »/9)+V(-»7)-0.
ESCBIBED CIRCLE. 285
The other forms are inapplicable, "because they would
itroduce impossible expressions. We have then to deter-
dne the values of Z, m, and n. If we put a — in (1), we
btain — = — ; thus — is the ratio of the perpendiculars drawn
7 m m r r
d the sides /3 = 0, 7 = 0, respectively, from the point where
he circle meets the line a = 0. Let r be the radius of the
ircle ; then we know from geometry that the perpendicular
rom this point on fi = is
r cot — sin C or 2r cos 8 — ;
similar expression holds for the perpendicular on 7 = 0.
ience
COS ~
n 2
m 9 B
cos —
2
2 A
I C08 2
Similarly - = x .
COS 4 -
Hence the required equation is
cos — V a + c°s — \/^8 + cos — \Ay = 0.
A A &
315. To find the equation to the circle which touches one
ide of a triangle and the other two sides produced.
Let the circle be required to touch the side opposite to
he angle A and the other two sides produced. Suppose
he origin within the triangle ; then for all points comprised
between the side a = and the other sides produced, a is
positive and ft and 7 are negative. Hence by Art. 309, the
6rm of the equation to the circle must be
V(- k) + V(™£) + V(wy) = 0.
286
ESCRIBED CIRCLE.
Hence, as before, by considering the point where the circle
meets the line a = 0, we have
n
m
-7T— G • • O
cos — - — sin 1 —-
2 2
cos sin — '
2 2
and
I
n
cos —
2
cos'-
COS
7T
^7;— rn?'
sin —
2
Hence the required equation is
A B C
cos -- sf (- a) + sin -V/?+ sin — Vy = 0.
Similarly the equations to the other two circles may be
written down.
316. The results in Arts. 306 and 312 which hold for
any conic section, will of course hold for a circle inscribed in,
or described about, a triangle respectively. We have only to
use the values of I, m y n, found in Arts. 313 — 315.
317. Let there be any quadrilateral, and let its sides be
represented by the equations
* = 0, w = 0, v = 0, t0 = O,
then the equation tu -f how = 0,
where Jc is a constant, represents a conic section circumscribing
the quadrilateral. For the equation represents a conic section
passing through the four points determined respectively by
t = and v = 0, t = and w = 0,
u *= and v = 0, u = and w — 0.
Also by giving a suitable value to k, the equation may be
made to represent any conic section passing through tnese
four points.
EQUATION TO A CONIC SECTION. 287
The above equation lias the following geometrical inter-
relation. If any quadrilateral figure he inscribed in a conic
sction, the product of the perpendiculars drawn from any
oint of the curve on two opposite aides bears a constant ratio
> the product of the perpendiculars on the other two sides.
We may observe that the terra quadrilateral is often used
i analytical geometry in a wider sense than in ordinary
ynthctical geometry. Thus, if we have four given points, we
»ay obtain three different quadrilaterals by connecting these
oiuts in different ways. Take, for example, the figure in
Urt. 76 ; and let A, B, C, D be the given points. The three
itferent quadrilaterals are (1) the figure hounded by AB,
* C, CD, DA ; (2)_ the figure bounded by A C, CD, DB, BA ;
*"liich in fact consists of the two triangles GAB and GGD;
1) the figure bounded by AC, CB, BD, DA, which in fact
insists of the two triangles GBC and. GDA.
Similarly, four given straight lines may be considered to
irm three different quadrilaterals by their intersections,
^ake, for example, the figure in Act. 76, and let the given
Lnes he EDO, EAB, AGC, BGD. The three different quad-
ilatcrals are (1) the figure hounded by GC, CE, EB, BG ■
2) the figure bounded by GD, DE, EA, AG; (3) the figure
Munded by AC, CD, DB, BA.
If four lines have for their equations
( = 0, u = 0, v = 0, io = 0,
he conic sections passing through the angular points of the
hree different quadrilaterals which these lines form, may be
Lenoted by the equations
tu + k l vw = 0, tv + k t uw = 0, tio+k t uv = 0.
M
318.
"We shall next consider the equation
uv + Jew* = 0.
This represents a conic section which passes through the point
letermined by u = and w = 0, and also through the point
letermined by v = and w = 0. Also each of the lines w =
md v = touches the conic section where it meets it ; for if
ve combine w = 9 with the above equation, we see that w =
288 EQUATION TO A CONIC SECTION.
also, that is, the line u = 0, meets the curve in only one point,
namely, that point in which u = and w = intersect. Dffii
larly the line v = touches the curve. Thus u = andt>=(_
represent two tangents to the conic section, and w = iejtt»
sents the corresponding chord of contact.
We may also shew in the following way that the line «=0
cannot cut the curve ; for points on one side of the line u=H
the expression u is positive, and for points on the other ail
of the line, negative ; but kw % is of invariable sign ; thusu=l
cannot cut the curve.
The geometrical interpretation of the above equation is i
follows. The product of the perpendiculars from any point i
a conic section on a pair of tangents bears a constant ratio
the square of the perpendicular from the same point on
chord of contact.
319. Next take the equation
Fu* + j»V = nV.
This may 1* written
(nw + mv) (nw — mv) = Pu*.
Hence by the preceding article
nw + mv = and nw — mv =
are tangents to the conic section represented by the equation,
and w = is the equation to the corresponding chord of con-
tact. Since these two tangents meet in the point of intersec-
tion of v = and w = 0, it follows that this point is the fd
of u = 0.
Similarly we may write the equation in the form
(nw + lu) (nw — lu) = «jV,
and infer that the point of intersection of w = and M = 0i
the pole of v = 0.
Hence it follows that the point of intersection of w=0ani
v = is the pole of w = 0. See Art. 291,
320. The following is a particular case of the precedinj
article
a* + /3" = «V. (See Art. 73.)
CONIC SECTIONS IN CONTACT. 289
Lppose the lines a = 0, j8 = 0, at right angles; then a' + ^S*
the square of the distance of the point (x, y) from the inter-
stion of a = and ft = 0. Hence the above equation repre*.
ats a conic section which has 7 = for its directrix, and the
bersection of a = and fi = for its focus. The lines
ny — a = and ny + a =
e tangents to the conic section touching it at the extremities
the focal chord # = ; also these tangents meet in the line
= 0; hence, the tangents at the extremities of any focal chord
zet in the corresponding directrix. Also the above tangents
eet on the line a = 0, which by supposition is perpendicular
j8=0; hence, the line which joins the focus to the intersection
'tangents at the extremities of a focal chord is perpendicular
that focal chord.
321. If u = and v = be the equations to two conic
ctions which meet in four points, then u+ Iv = will repre-
nt any conic section which passes through the four points
" intersection. This will be obvious after the proofs given
"similar propositions.
Also if to = and w = be the equations to two straight
nes, u + Iww = will represent any conic section passing
irough the four points in which the lines w = and w =
«et the conic section u = 0.
Also u + lw 2 = will represent a conic section passing
rough the points of intersection of the conic section u = 0,
id the line 10 = 0. This conic section will have the same
ngent as u = at the points where u = and w = inter-
ct; we might anticipate this would be the case from ob-
rving the interpretation of the equation u + Iww = 0, and
pposing the line w = to approach the line w = 0, and
timately to coincide with it. We may prove it strictly
r taking one of the points where u = meets w = for the
igin, and the line w = for the axis of x ; thus u becomes
tne form
Ax* + Bxy + Cy* + Dx + Ey ,
id we can see, by Art. 283, that
Ax* + JBxy + Cy* + Dx + ]Sy =
T. C. S. 19
290 CONIC SECTIONS IN CONTACT.
and Aof + Bxy + Cy* + Dx + JEy + ly* =
have the same tangent at the origin.
Also by giving a suitable value to I the equation u\ Itf
may be made to represent the two straight lines which
the conic section u = at the points where it intersects
straight line w = 0. This may be inferred from Art. 29>J
the equation w = is equivalent to the equation
hi- 1 - '
and the equation u = is equivalent to
(M-'J
Thus by taking I = — 1 we have u + Iw* = yacy ; and
equation xy = denotes the two tangents to the conic
u = at its points of intersection with the straight line w s
322. Pascal's Theorem. The three intersections 0/A
opposite sides of any hexagon inscribed in a conic section <A
in one straight line.
Let r = 0, * = 0, * = 0, w = 0, v = 0, t0 = O,
be the equations to the sides of a hexagon which is inscribed
in the conic section #=0. Let the hexagon be divided by 1
new line 6 = into two quadrilaterals, one of which hash
its sides tne lines obtained by equating to zero successive^
r, s, t, <p, and the other the lines obtained by equating to »#
successively, u, v, 10, <£. Now we know that if o, J, I, *»**
appropriate constants, the equation to the conic section flty
be written in the forms
as<f> + brt = and lv<f> + muw = ;
therefore as<j> + brt and lv<j> + muw must each be identic*
with 8; therefore
as(f> + brt = f lv<f> + muw ;
.\ (as — lv)<f> = muw — brt.
Ol.
I;.'
pascal's theorem. 291
The right-hand member of this equation vanishes when
a.nd r simultaneously vanish, and when u and t simulta-
>nsly vanish ; also when w and r simultaneously vanish,
i when w and t simultaneously vanish. Since the left-
id member is identically equal to the right-hand, the left-
id member must also vanish in these four cases ; that is,
* of its two factors <j> and as — lv must vanish in each of
se four cases. By construction, <j> = represents the line
ling the point determined by r = and w = 0, with the
nt determined by t = and u = ; and thus we see that
- lv = is the line joining the intersection of u = and
with that of t = and w = 0. But the line as — lv =■
iously passes through the intersection of a = and v = ;
refore the three points determined respectively by
t* = 0andr = 0, £ = Oandt0 = O, * = 0andv = 0,
on a straight line.
It is to be observed that if six points be connected by
slight lines in different ways, as many as sixty figures can
formed which may be called hexagons in an extended sense
that word. Thus for six given points on a conic section
jre will be sixty applications of Pascal's Theorem.
323. Let s = be the equation to a conic section, and
w = 0, v = 0, w = 0,
lations to three straight lines ; then
►resent curves of the second degree touching the proposed
dc section. By properly choosing u y v, w, I, m, n, we may
ke each of the last three equations represent a pair of
slight lines touching s = 0. (See Art. 321.) Tnus, if
re be a hexagon circumscribed round the conic section
: 0, the equations
9 _ZV = 0...(1), s-wV = 0...(2), s-nW = 0...(3),
y be taken to represent the six sides of the hexagon.
19—2
292 brianchon's theorem.
By combining (1) and (2) we obtain
8 — I V — (* — mV) = 0, or (mv — lu) (mv + lu) = 0...(4),
for the equation to a pair of lines which pass through tl
intersections of (1) and (2).
Similarly (nw — mv) (nw + mv) = Q ..(5)
represents a pair of lines which pass through the intersectioi
of (2) and (3). And
(Zw — nw) (lu + nw) = (6)
represents a pair of lines which pass through the intersectio]
of (3) and (1).
The six lines which we have obtained may be arrange
in four groups, each containing three lines which meet in
point, namely,
mv — lu = 0, nw — mv = 0, lu — nw = 0,
mv + lu = 0, nw -f mv = 0, lu — nw = 0,
mv + lu = 0, nw — mv = 0, lu + nw = 0,
mv — lu = Q 9 nw + mv = 0, lu + nw = 0.
This result is consistent with Brianchon's theorem; \
hexagon be described about a conic section the three diagffl
which Join opposite angles meet in a point.
For suppose that a hexagon is described round a cc
section, and let its angular points be denoted by A, B, 0,
JE, F. By properly choosing u, v, w, I, m, n, we may a
equation (1) denote the lines AB and DJfi, equation (2) dei
the lines BG and EF, and equation (3) denote the lines
and FA. We will now examine what lines are detenu
by equations (4), (5), and (6). Equation (4) determines
two lines which pass through the intersections of the !
determined by (1) and (2) ; and as the signs of I and m
at present in our power we may take them so that mv — It
shall represent the line BE, and then mv + lu = will n
sent the line joining the point which is common to AB
brianchon's theorem. 293
^with the point which is common to BC and DE. Simi-
"ly as the sign of n is still in our power, we may take it
that nw — mv = shall represent the line CF, and then %
>+mv = will represent the line joining the point which
common to BC and FA with the point which is common
CD and EF. One of the two lines represented by (6) is
D, and the other is the line joining the point wnich is
Qmon to DE and FA with the point which is common
CD and AB; it is however not obvious how we are in
neral to discriminate between these two lines. Thus the
oof of Brianchon's theorem is not perfectly satisfactory,
d accordingly we shall give another proof by which the
eorem is deduced from that of Pascal.
Let the angular points of the hexagon be denoted as
fore by the letters A, B, C, D, E, F. Let the line be drawn
[rich passes through the points of contact of the conic sec-
m and the tangents AB, BC; also let the line be drawn
hich passes through the points of contact of the conic sec-
)n and the tangents DE, EF; and let P denote the point
hich is common to these two lines. Then P is the pole of
E\ see Arts. 103, 120, 289. In the same way we may
itermine the pole of CF which we shall denote by Q, and
e pole of AD which we shall denote by B. By Pascal's
eorem P, Q, and B lie in a straight line ; hence OF, BE,
id AD meet in a point, namely, in the pole of the line
QR; see Art. 291.
For further information on the subject^ this chapter the
ident is referred to Salmon's Conic Sections.
EXAMPLES.
1. Shew that if a — c : a' — c' :: b : V, a circle may be
iscribed through the intersections of the two conic sections
as? + bxy + cy 2 + dx + ey + jf = 0,
a'x* + Vxy + c'y 2 + d'x + e'y +/' = 0.
Find also the condition that a parabola may be described
issing through the origin and the points of intersection of
ese curves.
2. Two conic sections have their principal axes at right
angles ; shew that a circle will pass through their points cf
intersection.
3. The equations to two conic sections are
Ay" + 2Bxy + Cx*+2A'x = 0,
af+ 2bxy + ex* + 2a x = 0.
Shew that the lines joining the origin with their points of
intersection will be perpendicular to each other if
a'(A+0)=A'(a + c).
4. An ellipse is described so as to touch the asymptote
of an hyperbola ; shew that two of the chords joining tbi
points of intersection of the ellipse and hyperbola are parallel.
5. If ct# = c 3 be the equation to an hyperbola (Art. 73),
then a^ — 0, a a — ^ = 0, a s — »*/9* = 0, are the respective eq»
tions to the asymptotes, the axes, and a pair of conjugate
diameters, n being any constant.
6. The straight lines which bisect the angles of a triangle,
meet the opposite sides in the points P, Q, M, respectivelT)
find the equation to an ellipse described so as to touch tk
sides of the triangle in these points.
7. From any point two straight lines are drawn, one in-
clined at an angle a, the other at an angle -^+a, to the «W
of a parabola ; shew that another parabola may be described
which, shall pas3 through the four point3 of intersection,
whose axis is inclined at an angle 2a to that of the given
parabola.
Prove that the equation to the conic section whicl
through the point (h, k), and touches the paraboli
y'^lx at the vertex and at an extremity of the latus rec-
tum, is
(y-£c)(i-2A) , = (y-2a:) s {^-M).
Shew that it is an ellipse or hyperbola according as tbe
point (h, k) is within or without the parabola.
9. A conic section touches tlie sides of a triangle ABC in
e points a,b,c; and the straight lines Aa, Bb, Co, intersect
.e conic in a, b', o; shew that
(1) the lines Aa, Bb, Ce pass respectively through the
Mersections of Be and Cb', Ca and Ac', AV and Ba,
(2) the intersections of the lines ah and a'b', be and b'c',
cand a'e', lie respectively in AB, BC, CA.
10. A conic section is described round a triangle ABC ;
neB bisecting the angles of this triangle meet the conic in
ae points A', B', C, respectively; express the equations to
l'B,A'C,A'B'.
11. If a conic section be described about any triangle, and
ie points where the lines bisecting the angles of the triangle
leet the conic be joined, the intersection of the sides of the
iangle so formed with the corresponding sides of the original
iangle lie in a straight line.
12. Interpret the equation
H(S
many parabolas can be drawn through four given
joints ?
3. If m=0, v = 0, w = represent the sides of a tri-
ugle, shew that the sides of any triangle which has one
le on each side of the former may be represented by
:re I, m, n are constants.
Find also the relation which must hold between I, m, «, in
order that the lines joining corresponding angles of the two
triangles may meet in a point.
14. A circle and a rectangular hyperbola intersect in four
loints, and one of their common chords is a diameter of the
ivperbola; shew that another of them is a diameter of the
circle.
15. ACA' is the major axis of an ellipse, P any point
the circle described on the major axis, AP, A'P meet t
ellipse in Q, Q'; shew that the equation to QQ' is
{a* + h*) y sin + 2h*x cos - 2a&* = 0,
the ellipse heing referred to its axes, and 6 being the a:
AC P.
If an ordinate to P meet QQ' in B, the locus of R i
ellipse.
16. The locus of a point such that the sum of the sqi
of the perpendiculars drawn from it to the sides of a j
triangle shall he constant, is an ellipse; and if the con
be so chosen that the ellipse may touch the side oppos
the angle A in D, then
CD : BD :: V : c\
17. With the notation of Art. 313, shew that the eqt
to the line through C and the centre of the circle is
18. Suppose in Art. 313 that D is the middle point
arc AB; then the equations to BD and AD are respecti'
o sin C + 7(sin.4-f-sini?) =0;
£ sin C + y{amA + BmB) =0.
19. In Art. 309, equation (4), if A', B", C be the ]
of contact of the triangle and conic section, shew tht
equation to A'B' is
hi + mv — nw — 0.
20. In the figure of Art. 292, suppose w = the eqi
to AC, u = the equation to BD, and w = the equati
EF, and that
ZV + mV-M s M> s =
represents a conic section passing through A, B, C, D\
express the equations to the tangents at A, B, C, D, an
EXAMPLES. 297
> the lines AB, BC, CD, DA. Shew also that the line FG
^8ses through the intersection of the tangents at A and B,
*id of those At G and D.
21. Find the condition that the line
\u + fiv + vw =
lay fowoA the conic section
*J(lu) + *J(mv) + *J(nw) — 0.
22. Give a geometrical interpretation of equation (1) in
Lrt, 304, and shew that it is a particular case of the theorem
i Art. 317.
23. Interpret the last equation in Art. 313 ; deduce the
>llowing theorem; if from any point of the circle which
ircumscribes a triangle, perpendiculars are drawn on the
ides of the triangle, the feet of the perpendiculars lie in one
traight line.
24. If ellipses be inscribed in a triangle each with one
)cus in a fixed straight line, the locus of the other focus is
conic section passing through the angular points of the
dangle.
25. Three conic sections are drawn touching respectively
ach pair of the sides of a triangle at the angular points where
hey meet the third side, and each passing through the centre
f the inscribed circle ; shew that the three tangents at their
ommon point meet the sides of the triangle which intersect
heir respective conies in three points lying in a straight line.
Jhew also that the common tangents to each pair of conies
atersect the sides of the triangle which touch the several
>airs of conies in the above three points.
26. With the angular points of a triangle ABC as centres,
,nd the sides as asymptotes, three hyperbolas are described,
iaving-4', U', C as their vertices respectively: prove that if
AA sin- = BB' sin J = CC sin-£,
29a EXAMPLES.
the intersections of each pair of hyperbolas lie on the axia of
the third.
27. The necessary and sufficient condition in order that
the equation fo 5 + mfcr + n y* = may represent a rectangnk
hyperbola is I + m + n = 0.
28. Shew that V(fa) + V(ntf) 4V(«7)=0 rei
general an ellipse, parabola, or hyperbola according as
tmn
fl m n\
\a b cl
is positive, zero, or negative ; where a, 6, c denote the lengths
of the sides of the triangle formed by a = 0, /3 = 0, y = 0.
29. Shew that Iffy + mya + naff = represents in genenl
an ellipse, parabola, or hyperbola according as
Pa 1 + m*J* + »V — 2lmab — 2mnhc — 2nlca
is negative, zero, or positive.
30. Find the condition that the line
Xu + fj.v + vw = O
may touch the conic section
lu 1 + mv' + nw 1 = 0.
31. Find the fourth point of intersection of the conic |
sections
Ivto + mwu + nuv = 0,
and I'vw + m'wu 4- n'uv = 0.
32. Shew that the equation to the radical axis of I
circles inscribed in a triangle and circumscribed about it ia
acosec A cos*^
c B cos' — + 7 c
t G =
33. Find the equation to the diameter of the ci
Iffy + mya + naff =
which passes through the point of intersection of the
/3 = and 7 = 0.
EXAMPLES. 299
4. Find the equation to the tangent to the curve
V(&0 + V(™£) + V(wy) = °>
h is parallel to the line 7=0; and thence shew that the
e of the curve is determined by
« = P = 7
mc + nb na+lc lb + ma "
5. From a point P two tangents are drawn to a conic
>n meeting it in the points M and N respectively ; the
through P which bisects the angle MPN meets the
I MN in Q ; any chord of the conic section is drawn
gh Q ; shew that the segments into which the chord is
ed by the point Q subtend equal angles at P.
t 300 )
CHAPTEE XVI.
SECTIONS OP A CONK ANHARMONIC RATIO AND HAR)
PENCIL.
Sections of a Cone.
324. We shall now shew that the curves whic
included under the name conic sections, can be obtain
the intersection of a cone and a plane.
Def. A cone is a solid figure described by the revc
of a right-angled triangle about one of the sides cont
the right angle, which remains fixed. The fixed 8
called the axis of the cone.
Let OH be the fixed side, and OHC the right-
triangle which revolves round OH. In order to ol
8ECT10NS OF A CONE. 301
3 such as is considered in ordinary synthetical geometry,
should take only a finite line OU; but in analytical
metry it is usual to suppose 00 indefinitely produced both
fs. A section of the cone made by a plane through OH
00 will meet the cone in a line OB, which is the
ition 00 would occupy after revolving half way round.
a section of the cone be made by a plane perpendicular
he plane BOO; let AJP be the section, A being the point
ire the cutting plane meets OC; we have to find the
are of this curve AP. Let a plane pass through any point
f the curve, and be perpendicular to the axis OH; this
ne will obviously meet the cone in a circle DPE, having
diameter DE in the plane BOG. Let MP be the line
wrhich the plane of this circle meets the plane section we
considering, M being in the line DE. Since each of the
nes which intersect in MP is perpendicular to the plane
)(7, MP is perpendicular to that plane, and therefore to
ay line in that plane.
Draw AF parallel to ED, and ML parallel to OB; join
1. Let AM=x, MP=y, OA = c, HOC=a, OAM=0;
angle AML will be equal to the inclination of AM to OB,
t is, to 7T — — 2a.
^ MP _ sin MAD __ sin Mn _xsm0
MA ~~ sin MDA cosa ' "~ cos a "
EM= FL =FA -AL = 2c aina- AL;
AL _ sin A ML _ sin (it — — 2a)
AM" sin ALM~ . fir \ '
sm^+aj
j T __ »sin_(^+2a)
~~ cos a '
rmr « • #sin(0+2a)
/. jEflf = 2c sin a * '- .
cosa
t, from a property of the circle, MP* = EM . MD ;
9 x sin ( . a? sin (0 + 2a)|
.\ #* = 42csina - -h
.cosa cosa J
302
SECTIONS OF A CONE.
If we compare this equation with that in Art. 282, mia I
that the section is an ellipse, hyperbola, or parabola, MW& |
sin 8 ain (8 + 2a) .
ing as j - is negative, positive, or zero, toil
+ 2a is less that tt, greater than ir, tt
is, according as
equal to tr.
Hence if AM is parallel to OB the section is a paiaholi,
if AM produced through M meets OB the section is an ellipse,
if AM produced through A meets OB produced through tk
section is an hyperbola.
If c = the section is a point if 8 + 2a is less than ir, tp
straight lines if 6 + 2a is greater than w, and one straight liw
if 8 + 2a = 7r. The section is also a straight line whatever
may be, if 6 = or ir.
The equation above obtained may be written
n {$ + 2a) (2c sin a cos a
is^a" [sm(0+2a) 3
-A;
suppose 8 + 2a to be less than -tt, so that the curve is ac
ellipse; theu by comparing this equation with the equation
y i = -5 (2a;c — a;*) , we have
2c sin a c
_ sin 8 sin (0 + 2a)
cos* a
Also
sin ($ + 2a)' sin (6 + 2a) '
_ ^1 - c 03 ' a — 1 B U1 ' (^+ «) ~ a" 1 * «} _ cos" (fl+a;
cos" a
cos* a
If we suppose in the figure of Art. 324 that AM is pro-
duced to meet the cone again in A', then 2a = A A ', as might
have been anticipated; also h may be shewn to be a mean
proportional between the perpendiculars from A and A on
the axis OH. Similar results may be obtained when the
curve is an '
ained when the
ANHAKMONIC BATIO.
303
Anharmonio Ratio and Harmonic Pencil.
15. We will now give a short account of anharmonic
: and harmonic pencils, which are often used in investi-
sj and enunciating properties of the conic sections.
et there be four straight lines meeting in a point ; then
l straight line ADCB be drawn across the system,
AB DB
AG' DC
will be a constant ratio.
uppose the point where the lines meet ; then
AB sin A OB
AO~ Bin ABO'
AO amAOO
AO~BmAOO''
AB sin A OB am ACQ
''* A ~ sin A C ' sin ABO '
. DB sin DOB sin D CO
mularly -BC~ amDOC b\d.DBO ]
AB DB sin A OB sin D OB
'"' AC*DC~sinAOC ' bjuDOC
304 HAKMO.VIC PENCIL.
Now suppose any other straight line A'DG'B' drawn
across the system, then since AOB and AOB' are the salt
angle, and so on for the other angles, wo have
AB DB _A'F D'B'
AC ■ DC A'C " DC"
which proves the proposition.
Similarly we can prove that each of the following is
constant ratio j
AB CB AC BC
AB ■ CD ana AD' BD-
32G. Defs. Any four Hues meeting in a point foni
pencil,
A straight line drawn across a pencil is called a tn
versal.
. , . .. AB BB AB t
Any one or the constant ratios — rr<^- vtt-v i ~rr. + -}
J AC DC AD (
AC BC.
AD ■ BD 1;
The pencil is called harmonic if AB.DC = AD.BC,
is, if the rectangle formed by the whole line (AB) and
middle part [DC) is equal to the rectangle of the other
parts {AD), {BC).
327. The harmonic pencil is so called because it dh
any transversal harmonically. For since AB.DC =
AB_BC
AD DC*
id, and t
that is, if we call AB, A C, AD, the first, second, a
quantities respectively, the first is to the third as the di
ence of the first and second is to the difference of the sei
and third.
When the pencil is harmonic one of the three cons
ratios of the pencil is equal to unity.
HABM0N1C PENCIL. 305
We shall sometimes select one of the anharmonic ratios of
* pencil, and confine our attention to it, and shall then speak
of the selected ratio as the anharmonic ratio of the pencil.
328. Suppose OA, OB, 00, OB form an harmonic pencil;
tf "We take any new* origin O, and join O'A, OB, O'U, O'B,
these four lines form a new harmonic pencil ; for the trans-
versal ABGB is cut harmonically.
329. The anharmonic ratio of a pencil is not altered if
the transversal meet the lines of the pencils produced, instead
of the lines themselves.
Suppose OA, OB, 00, OB to be a pencil, and let a
transversal A'B' C'B' meet three lines of the pencil, and the
fourth AO produced in A'. The angles A' OB', A OB are
supplemental ; and so are A OB, A' OU ; and so on. Hence
any anharmonic ratio formed on ABGB is equal to the cor-
responding ratio formed on A'B' C'B'.
330. Suppose AB. CB = AB.BC, so that OA, OB, 00,
OB form an harmonic pencil. By the last proposition
A'B' C'B 1 AB . CB
AD ' C'B'~ AB ' CD '
.\ OA', OB', OC, OB' form an harmonic pencil.
T. c. s. 20
306
HARMONIC PENCIL.
Similarly OG\ OB\ OA\ and DO produced thiongl
will form an harmonic pencil. Thus from one harm
pencil by producing the lines through the vertex, we
derive four other harmonic pencils.
331. The lines whose equations are a=0, j8=0, a— H
a + k/3 =5 form an harmonic pencil.
Let OM be the line a = 0,
ON /9 = 0,
OP a-£/9 = 0,
OQ a + &£ = 0.
Let a transversal meet the pencil in mpnq ; then (Art
sin PO M _, sin QOM
&inPON~ sin QON]
sin POM sin QON
"'• Bin PON* sin QOM~ ;
.-. (as in Art. 325)^.-^ = 1;
pn qm
•'. pm . qn —pn • qm.
HARMONTC PENCIL. 307
The same result will follow if we draw the transversal in a
lifferent position. The harmonic pencil is so formed that its
ntside lines are always one of the two a = and ft = 0, and
neof the two a — kfS=*0 and a + 4/8 = 0.
332. The anharmonic ratio of the four lines a=* 0, & = 0,
c-jty9 = 0, a + £'£ = 0, is«.
For as in the preceding article we have
sin POM . ainQOM j,
sinPOtf"*' sin QON~ k;
k
herefore, by Art. 326, jj expresses the anharmonic ratio.
333. Article 331 will also hold if the equations to the
ines be u = 0, v = 0, u — kv = 0, and u + kv = 0. For, by
irt. 57, we have u = \a, v = fi/3, where X and fi are constant
quantities ; hence the equations u — kv = and u + kv =
nay be written \a — kfi/3 = and Xa + &A&i8 = 0, or a — k'/3 =
ind a + A'yS = 0, where k f =~. Hence Article 331 becomes
mmediately applicable.
334. The four lines EB, EC, EG, EF, in Art. 76, form
in harmonic pencil ; for their equations are
u = 0, 10 = 0, lu — nw = 0, Zw + wi0 = O.
By symmetry FB, FA, FG, FE, will also form an harmo-
lic pencil.
Also GD, GO, GF, GE form m harmonic pencil, for their
equations are respectively
lu — mv = 0, mv — 7M0 = O, Z# — wit;- (mv— nw) =0,
lu — #iv + mv — nw ss 0.
335. A straight line drawn through the intersection of two
angents to a conic section is divided harmonically by the
;urve and the chord of contact.
7ft— a
308 HARMONIC PENCIL.
Eefer the curve to the tangents as axes ; its equation will
be of the form (Art. 293)
S + l- i ) ,+ ' M8f " (1) -
Suppose a straight line drawn through the origin, and let
its equation be (Art. 27)
?-£ »
Thus the distances from the origin of the points of inter-
section of (1) and (2) will be the values of r found from the
equation
(i/f ItiT \
I . m 1\ 2
or
If r' and r" be the roots of the equation, we have
?+?-«(M) »
Also the equation to the chord of contact is
hi- 1 - 0-
Hence for the distance (r x ) of the point of intersection of
(2) and (5) from the origin, we have the equation
A + X = 1 ' "f.-x + i (6) -
From (4) and (6) we have
r x r r
thus r x is an harmonic mean between r' and r".
PBOPERTIES OP A CONIC SECTION. 309
Since LMNO is divided harmonically, if from any point in
AS we draw lines to L, N, and 0, these lines with An form
an harmonic pencil. A particular case is that in which the
point in AB is the intersection of the tangents at N and L,
which we know will meet on AB. (See Arts. 103, 186.)
336. Let A, B, C, D be four points on a conic section,
and P any fifth point. Let a denote the perpendicular from
P on AB, @ the perpendicular from the same point on BC,
7 on CD, 8 on DA. Then by Art. 317 we know that
wherever P may be, ay bears a constant ratio to f$8. Now
AB . a = twice the area of the triangle BAB
= PA.PB.&inAPB;
PA.PB.BinAPB
• % *~ AB
Similar values may be found for j8, 7, 8. Thus
PA.PB.PG.PD . ,™ . „ Dn
.- „„ sin APB . sin CPD
AB. CD
bears a constant ratio to
PA.PB.PG.PD . wn • t>t>a
> p ^ ATl sm BPC. sin DP A :'
BC.AD '
. sin APB . sin CPD . . . _, ^ . . .. ,
. • — — ^57r — s — T\T>A 1S constant, that is, the pencil drawn
sm BPC . sm DP A ' ' r
from any point P to the four points A, B 7 C } D } has a constant
anharmonic ratio.
( 310 )
ANSWERS TO THE EXAMPLES.
CHAPTER I.
8. The co-ordinates of D are ^(x x + x a ) and i^+yj. The |
co-ordinates of G are J (x x + x a + # 8 ) and | (t/ x + y a + y J.
10. Let r and be the polar co-ordinates of C. Then the
angled 0(7 = the angle £00 ; that is, 0-0=0,-0; .-. tf^M+fJ.
Again, from the known expression for the area of a triangle
(see Trig<mometiry, Chapter xvl), we have
triangle AOB = \r x r a sin (0 a - X ),
triangle A0C= $r x r sin (0 - X ),
triangle J50C = £ r 8 r sin (0 8 - 0).
Thus r^ sin (0 a - X ) = r,r sin (0 - X ) + r a r sin (0 a - 0)
= r(r 1 + r s ) sin £(0,-0,);
.-. r (r x + r a ) = 2r x r a cos £(0 S - 0J.
CHAPTER III.
1. (1) y + 2s = l. (2)« = 2. (3) y = a?. (4) « = 0.
2. 3,-4 = -3(aj-4), y -4 = J(aj-4).
3. y-l = (>/3-2)*, y-l=-(^3 + 2)ir.
4. y = x, y=-a>. 5. y = -7«a?, « = 0.
6. 90°, a?=-i, y = $. 7. 60°. 8. 45°.
9. y = *(aj-a). 10. y=». 11. 2J2.
ANSWERS TO THE EXAMPLES. 311
19 °*> -.o _ «& ni * y_l 1
15. (1) The origin. (2) Two straight lines, y = x and y = — x.
( 3) Two straight lines, a? = and a? + y = 0. (4) The axes. (5) Im-
possible. (6) Two straight lines, x = and y = a. 16. (1) Two
straight lines, x = a and y = b. (2) The point (a, 6). (3) The
point (0, a). 17. The lines y — x and y = 3cc 19. 4=y = 5x,
and 3y + 2a-20 = 0. 20. Let a be the length of the side of
the hexagon; the equations are to AB, y = 0; AG, y,j3 = x;
AD,y = x>jZ; AF,x=0; AF,y + xj3 = 0; BC,y = >jZ(x~a);
BD,x = a; BE, y + JZ(x - a) = ; BF, y J3 + x-a = ; CD,
y + xj$ = 2a ^3; CF, y J3 + x = 3a; CF, 2y = a J3 ; BF,
y=J3a; BF, y,j3-x=2a; FF, y-xJ3 = aJ3. 21. If
( x i> l/i)> ( x a> y«)» ( x z> ^a) ^ e *^ e an g u l ar points, the co-ordinates of
X "I" X 1/ -i- 1/
the point midway between the first and second are 1 * , y * y * ;
similarly the co-ordinates of the point midway between the second
and third points are known ; and then the required equation can
be found by Art. 35. 22. ^4 *** «■ 24. - +?= 1,
J m a -l a b '
- = r ; tangent of the angle between them — 5 — =7- . 29. The
a ° a — 6*
points whose absciss® are a + r J (a 2 + b 8 ) and a - j- J (a 9 + 6*).
31. ^~^ - 35. 90°. 36. ^(0) = Ogives a system
of lines through the origin; sin 30 - gives the three lines y = 0,
y = a; ^/3, y = — x J3. 40. The second pair of lines bisect the
angles included by the first pair. 44. Let ABC be the triangle ;
take A for the origin and lines through A parallel to the two
given lines as axes; let x x , y x be the co-ordinates of B } and x a , y a
those of C. Then it may be shewn that the equations to the three
diagonals mentioned are
y-y^t^^-^ *-*—?* *-*.—£
x
from these equations it may be shewn that the three diagonals
meet in a point 45. Take as origin and use polar equations
312
ANSWERS TO THE EXAMPLES.
to the given fixed straight lines. 4C. Let x s be the abscissa of tit
point of intersection of the two lines ; then, the area of the triangle I
in {(^-c^x,. 47. This may be solved by Art. 11. Or™
may use the result of the preceding question ; for by drawing n
figure we shall obtain three triangles to which tho precedinj:
question applies, and the required area is the difference betw
two of these triangles and the third. The result is
A;
('■-»,)' -
>(-,-J >h-»J/'
which may also be written thus
{», K -■".) + '■(■»,- "J <-c, (»,-■»,)}•
i(-,-»JC,-».H«,-«,)
That sign should he taken which gives a positive result.
CHAPTER IV.
aba b''
7. Since the required line is parallel to the line considered b
Example 5, we may itssunie for its equation
BO0B4-^O08it + 4-=fli
where h ia some constant to be determined. Now at the middle
point of AB, we have
— Si** -#-j""'j
therefore — = sin B cc
thus k is determined.
*B+k=Q;
I
13. (mn'-m'n)v, + (nl' ~n'[)v + (lm' -l'm)u
14. ah{u-v) + c(b + a)w=0.
15. Assume for the required equation la + m/3 + ny — ; a
centre of the inscribed circle a = ^3 = -y ; thus £ + to + ji=:0; at tie
centre of the circumscribed circle a, 8, y are proportional respec-
ANSWERS TO THE EXAMPLES. 313
jT to cos -4, cos B, cos G; thus I cos A + w cos 2? + w cos (7= 0.
5e the required result may be obtained.
8. ToCP, 2mv-nw=0; to DP, 2lu - 2irw + nw = ;
to AQ, lu — 2mv + 2nw- ; to 2?$, fo — 2mv = 0.
3. It may be shewn that ifu = 0, v = 0, w = denote the
of the triangle, the lines AP, BP, GP may be denoted by
• nw = 0, nw '— lu = 0, and lu — mv = respectively; then
quations to the other lines can easily be expressed.
4. Take a = 0, /? = 0, y = to represent the sides of the
gleA'B'C ; then the equations to BO, GA, AB will be respectively
' = 0, y + a = 0, a + /? = 0. Then the equation to AA! will be
f = 0, so that AA! is perpendicular to BO,
5. The equation to 0(7 is /? — y = ; take /? — y *- Act =
hie equation to the line drawn through JD. Then it will be
1 that the equation to OF is ft — y — A (a — y) = 0, and that
iquation to O'E is /?-y- A (a + /?) = 0. Thus at the point P
ave /3 = — y, The same relation holds at the point Q.
CHAPTER Y.
. (a? + yy = a'(a?-y>). 3. y"=cJ2a<-£.
. y"sin s a=4aaj / . 6. By Art. 83, we have
sin (<o — a) sin (a> — 5) , sin a . sin 8
m = — > L n - J HL m'=- — , n=-r— -.
sino> 8ino> sin a) sino>
CHAPTER VI.
(1) Co-ordinates of the centre 2 and — 2, radius 3.
(2) Co-ordinates of the centre — 3 and f , radius J.
The first line meets the circle at the points (— 4, 3) and
4) ; the second at the points (0, — 5) and (- 5, 0) ; the third
es it at the point (— 4, — 3).
For determining the abscissae of the points of intersection
314 ANSWERS TO THE EXAMPLES.
we have x'fl + fy + ?£ $-&)*- 2ax + W-3b& = 0; if the list
touches the circle vc must have (kb — /«*)'+ 2AA(fan-Ai)=W.
9. 2y + 3s = 0. 14. g? + y'-xy -ke-Jy-0.
15. Inclination of axes 120°; co-ordinates of the centre eacb-I,
radius = A. 16. Inclination of axes 60°; co-ordinates of tit
18. a? + y* + xy + x + y-\ = §.
u *♦•-.(.♦#.„-
28. TJae the equation
co-ordinates, we have
r + J(r' + a' - 2ra cos $)=/& + a'- 2ra «*(Z-o)]>
reduce and we get l t /3r~2aco8. (& - ~}\ = 0; thus the Jocui
tlie circle circumscribing the triangle.
30. sin'a + sin'/3 + sin , r +... = coa' a + cos* £ + cob 1
and sin 2a + sin 3/3 + sin 2 r + ... =0.
32. If the perpendiculars are both on the same side of theliM
the locus is a circle ; if oil different sides the locus consists
straight lines. 33. A circle. 34. A circle. 36.
the quadratic in r ; it will be found that r = 2a coa 9 or — a
thus the locus consists of a straight line and a circle.
38. Take the extremity of the diameter as the pole; it wJ
follow from Example 37) that the tangent at P is represented !j
the equation 2c cos' a = r cos {2a — 9), and the tangent at Q by ti'
equation 2c cos' /3 = r cos (2/3 - 9). These tangents meet at T,*
at that point we have
from this we shall find tan 9
ANSWERS TO THE EXAMPLES. 315
bo that if G be the centre of the circle Ct = s tt — -*- •
2 cos p cos a
Cence we can shew that Cq-Ct = Ct-Cp.
CHAPTER VII.
4 j 3a5
. a; = y, and as + y = .
a + o
5. Let y=mx be the equation to one line ; then
db is a quadratic for finding tw, and we may replace m by - .
CHAPTER VIII.
1. y=2x. 2. y*=5ax — x*. 3. The locus consists
T two parabolas of which the centre of the circle is the common
jots, and the directrices are the two tangents to the circle which
Pe parallel to the fixed diameter. 4. The second curve is
parabola having its axis coinciding with the negative part of
le axis of y ; the curves intersect at the origin and at the point
«=4a, y= — 4a. 5. y = x + a. 6. tan~ l £. 7. y + a?=3a.
. At the point (9a, - 6a) ; length 8a J2. 9. y = 2a J '3,
= 3a. 11. The abscissa of the required point is or 3a.
J. The curve is a parabola having its axis parallel to that of y,
id its vertex at the point x = £, y = £. The line is a tangent at
ie point 05=1, y = 0. 20. Abscissa of required point is
. Q^. + A\ ordinate -(^. + fA; length of chord j % (4a*+ f)*'
i. Locus of #, a; = - 2a. Locus of @, X s = ay 2 . 23. Refer
lo parabola to P5 7 and the diameter at P as axes. See Art 151.
J. Bee Art. 155. 27. Transform equation (1) of Art. 125
\ polar co-ordinates, and we shall deduce r = 2a ^g ..
ANSWERS TO THE EXAMPLES. 317
V — J&
= 7 , where (a?, y) is any point on the tangent ; thus
* — _« & = — — t-^ ; this will give the first form of the equation.
TPlie second form maybe deduced from the first ; the student will see
Ifcereafter what suggested the second form ; see Arts. 321 and 322.
tf€L The equation y*= 4<we represents the parabola ; and the equation
^f— 2ax = 2ah represents the chord of contact; hence the equation
(ky — 2ax) = 2atyf represents some locus passing through the
tion of the parabola and chord ; then see Art. 61 .
CHAPTER IX.
1. -75. 2. y + ex = a; the intercept on the axis of
5 f a> ... . . , . o
t
: ; '^= — ; and the intercept on the axis of y = a.
r - ^m» The excentricity is determined by e 4 + e 9 = 1 . 5. y = - (as + a) ;
3&#=— ; the lines are parallel if 2e*=l. 6. y = ~(x-ae);
are ~ ae
the abscissa of the point of intersection is -= 5 .
* 1 +e
» /i \/ \ x -l 1 o 2a*e-ax'(l+e*)
■ 7. y=-(l +«)(*-«); tan' TTiT7 . 8. ^^Ij -
a 9 b*
9. The co-ordinates of the point are x = —tt-s — t*\ > V = -77-* — ra\ •
a b
10. The co-ordinates of the point are x = -t~ , y = -75 .
19. It will be found that the circle falls entirely without the
ellipse if the inclination of the two parallel straight lines to the
ae x 11
major axis be greater than tan" 1 -7- . 22. - cos <£ + -7 sin <£ = 1.
25. The co-ordinates of the required point are as = -^ J-,
o — ae
t/ = —-i 5-^: the lines are parallel when e 4 +e 8 =l.
28. x* + y* -x(ae + xf)-yy' + aex'=0. 30. Ifthepoint(A, k)be
ANSWERS TO THE EXAMPLES.
between the directrices, the sum of the perpendiculars is -,, t , , „.r.
if the point (h, k) be not between the directrices, the sum of the pa-
, the upper or lower sign being tabs
+ b'k")
according as h is positive or negative. 31. A circle baviagi
centre at the centre of the ellipse and radius —a+b.
32. y^^x^^n'+b*). SeeArt. 171. 34. Locnsis*
circle fc' + y^ra' + t'; this may be deduced from the second [W
of example 33. 33. See remark on Ex. 55 of Chap, rat
-12. The first part of this example may be solved by finding tk
equation to the line passing through the points of intersection
the two ellipses. 45. x , + y'=(a'+b') i (x + y). *& ^ i*
a, k be the co-ordinates of an external point ; the eiyiaM
to the corresponding chord of contact ia a'Uy -^ b a itx -
equation to the lino through (A, k) perpendicular to the
(y-i)6*A=«'i(as-A). We require that the latter tine skll t<
a tangent to the ellipse ; the necessary condition may be found ty
comparing this equation with the equation y = rnx + ^/(niV^-f^l
tlms we shall obtain for the condition k'a" + 7t a 6° = /(*&* (rf-™'
48. a , (y + 2y*) + 6*(:e , + 2*A) = 0. 52. An ellipse. 33. Tk»
locus is an ellipse ; if A be the origin, AB the *•*'* of x, each of tlw
co-ordinates of the focus is equal to half the radius of the cirdc
54, -p- . 55. Put acos£ for x and & sin ^ for y in til '
preceding result (Art. 163); then the greatest value is ■$.■
57. Let P denote a point on the ellipse, and Q the centit
of the circle inscribed in the triangle SPH ; then if y" be the
ordinate of P it may lie shewn that the radius of the circle *H<&
area i>f trian gle SPH ey" ,. , .
" seiuiperimetor of triangle = TTi '' *"■ M ™° ordinat * rf •
Let x" be tie absciss* of P, then it may be shewn that the ••
eoiss* of Q is ex' ; thus it will be found that the required locus i
an ellipse. fig. Find the point in which SZ meets the noma
•t P \ also find the point in which HZ' meets the non
it vtiU then *.^eu \iis& VJ» -yiav\s, esouada.
ANSWERS TO THE EXAMPLES. 319
CHAPTER X
1 . xb (bx' — ay") + ya (at/ + bx*) = a*b*. 2. Refer the ellipse
tx> the diameter and its conjugate as axes. 3. See Art. 11.
S. r (a*sin a 6 + b* cos* 6) = 2ab* cos 0. 9 and 10. Use the re-
sult of 8. 12. Result the same as that in Ex. 11. 13. They
intersect when = and when = 5 • 14* The equations to
-ibe tangents at the ends of the latera recta are (Art. 205)
r (ecos 6 + sin 0) = a (1 - e*) ; r (sin - e cos 6) = a (1 + e f );
r (e cos 0- sin 0) = a (1 - e*) ; r (sin + e cos 0) = - a (1 + e 2 ).
e equations to the tangents at the ends of the minor axis are
mjx0 = b; rsin0 = -&. 15. A straight line through S.
Art. 205. 17. cos 6 = - ^^7, r = a (1 + ee\ 18. Be-
h n
'tween - and r . 20. See Art. 208. 22. The sine of the
a
angle between the radius vector from the centre and the tangent
m — , where p*(a* + 6 s - r*) = a*b* by Art. 196 ; then the least value
P*
d^j may be shewn to be when 2r 2 =a 2 +6 a . 29. It may be
shewn that the axis of the parabola must coincide with one
of the axes of the ellipse, hence the latus rectum will be either
2a* 2b*
tth — Ta\ or ■ // fl — t5v • 31. An ellipse. 32. An ellipse.
35. XJse the polar equations to PQ and pq ; see Art. 205.
38. Two of the sides of the parallelogram are determined by the
X 11
equations - cos <£ + j- sin <£ = ± 1, and the other two by the
x 11
equations - cos ^' + r sin<£' = «fel ; see example 22 of Chap. n.
It may be shewn that the diagonals of the parallelogram inter-
sect at the centre of the ellipse ; then if the centre of the ellipse
be joined with two adjacent corners of the parallelogram the
triangle thus formed is one fourth of the parallelogram; and
ANSWERS TO THE EXAMPLES.
of the triangle ia known by example 7 of Chap.
The abscissa is , a]l , and the ordinate — . 42. Th»
b a
co-ordinates of the intersection of the tangents are
F.f 41 • call them h and k, then use the second, form given tt
Ex. 35 of Chap. ix. 44. The greatest value may be found tj
substituting for a/ and y' their values from Art. 168; it b
ab (J2 — 1). 47. An ellipse. 48. An ellipse referred tt
its equal conjugate diameters. fil. This may be solved bj
means of Ex. 50. Or we may take the usual axes, then if x, f i
the co-ordinates ol'J' tlu>sn of J/ will Vns- -,—tt - and ^ t y r'i,
a +b
those of if will b,^i'«a*j!. H„»
tion can be completed.
CHAPTER XI.
1. y' — 3a;* = — 3a'. 2. A straight line.
CHAPTER XII.
3. Let a line be drawn through the focus meeting the hyperWi
in P and p and the asymptotes in Q and q ; then it may he shei*
required length is half the difference of .Pp and ^ff- 4- Tab
the centre of the circle as the origin, AB as the axis of
diameter parallel to PQ as the axis of y ; then the local
by the equation ;/'-x'-a', and is therefore a rectangular
referred to conjugate diameters. 9. By example 53 of Chapter B
we shall obtain tan a = ^ ( ' ~ ^ ■ ; .: (k + a)' ten' a = k' - id
.: (h + a)'sec'a = ^ + (!i-a)'. 10. Both the diameters nm*
•meet the curve ; it will be found that this requires the ccnjagi*
axis to be greater than the transverse axis.
ANSWERS TO THE EXAMPLES. 321
CHAPTER XIII.
1. The equation may be written (x - 2y) {x—2y — 2a) = 0, and
therefore represents two parallel straight lines ; a line parallel to
them, and midway between them, will be a line of centres.
b c
2. ,A=«, & = q. 3. Two parallel straight lines. 4. A
parabola. 5. An hyperbola if the angle A is less than ^,
fcn ellipse if it is greater than -= , a straight line if it is equal to - .
5. The equation to the hyperbola is a'y* = a*b 2 — iab'x + 36V; the
Asymptotes are determined by the equations ay = ±(x — -~- )b J '3.
3. The locus is then a straight line which coincides with the
saraalaxes. 10. Use Art. 205. 11. ^—. 13. Tan" 1 ^.
4 6
14. {ay + xJ(PlT)y-2apP'x-a*(P + P)y + a 9 pp'=0-
17. (1) A circle about the other focus of the given ellipse as
centre; (2) an ellipse about the other focus of the given ellipse
as focus, and having the same excentricity as the given ellipse.
18. The equation is (y- 3a? + 1) (y — 2x + 4) = 0, and therefore
represents two straight lines. 24. Use the result given in the
last example to Chap. vm. 26. The equation may be written
(x* + y* + xyj2 -a 8 ) (x a +tf-xy ^2- a 8 ) = 0.
CHAPTEE XIV.
2. Each locus is an ellipse. 4, 5, 6. Use the equation
x* tf
in Art 294. 7. The equation to the ellipse is -5 + v« = 1 ;
the equation to the chord of contact is — « + ts- = 1 ; hence the
equation ^ + ^ - — + j? represents some locus passing through
the points of contact 10. The equation to the hyperbola
T. C. S. 21
322 ANSWEK9 TO THE EXAMPLES.
ia (y — k) b'x = {x — h) a'y. 1 2. Let y 1 , y" denote the two onli-
nates wLicli correspond to the same abscissa of; then
^= - brf + V(4V- «*" +/). V" = ~ M- ^(SV- afl^+ZV
The equations to the normals are, by Art. 284,
{y - y) (ax" + bf) = ty + laf) (x - a"), and
(y-y")(atf+by")=(y" + baO(x-«0;
by addition (o -J*)^ (y + 2i«') +6/=0 ... (1);
by subtraction 6 (y + 6a/) -{a-l')x' = x—x',
therefore x~ (1 + 2b'- a)=x-by (2).
Substitute the value of x 1 from (2) in (1) and the required equip
tioii will be obtained. The locus is an hyperbola. 13. Loan
& conic section, which passes through H and R, and through tht
intersection of the fixed lines. 18. A circle having its centre
on the line joining the two poiots. 19, Two loci, au ellipte.
and & parabola. 20. A circle. 23. See Art. 295
26. TJso the equation to the parabola given in Art. 294, and
the equation to the circle given in Example 21 to Chap, ti
29. j-ain29 = c. 30. *V + ff*** = «*• 32 - See Exampfc
30 to Chap. x. 35. An ellipse. 37. In the first case the
locus is a circle; in the second it is a straight line. 38. A ci
having its centre at H. it. — r + £ = l. 46. The ei
tion is y' = ia(x-Ba). 50. The line --^=0, bisects the
chord of contact, and is therefore parallel to the axis of the par*-
bola; if through the point (a, 0) a line ho drawn making the
angle with the tangent at that point as the axis mokes, the fucm
must ho in this line ; y (a + 2t cos w) + b (x — a) — is the equation
to this line. Similarly wo can draw a line through the pqifi!
(0, 5) which will also contain the focus. 52. We may bkt fcl
the equation to one normal y — mx — am — am\ and for the
x = m'y— am' -am 13 ; a\aa m'=-m. Then by addition y+<c= p
Substitute for jii in the first equation and reduce ; ttrns we obti
2a(x + y)-(x-y)'. 53. We have to eliminate m babnfl
y-mx = -
JV + m-bV -"">"- J^W+V)-
ANSWERS TO THE EXAMPLES. 323
Square and add ; we shall obtain after reduction
^+«=— 7 (I ' (!)•
Also (y - wmc)* (a* + rnfV) = (my + a?)* (mV+ J") ;
by reduction we obtain
(«y-6V)(m-I) = -2^(aV6«) (2).
From (1) and (2)
(a 8 + V) (a 9 + y") (a Y + 6 V)* = (a 8 - bj (a Y- 6V) 1 .
54. Suppose the figure in Art 192 to represent the ellipse
and the conjugate diameters. Take the equation in Example 23
of Chapter ix. for the equation to the normal at P, and an ana-
logous equation for the normal at D. Let Q denote the point of
I intersection of these normals, and x, y its co-ordinates. Then it
y 'will be found that
ax = (a*— b*) sin <f> cos <£ (sin <f> — cos <f>),
by = (6* — a 8 ) sin <f> cos <£ (sin <f> + cos <£).
Similarly we can determine the co-ordinates of the point of inter-
section of the normals at P and D; denote this point by E. Then
express the area of the triangle CPE, which is one-fourth of the
required area.
55. Take the centre of the square as the origin, and the axes
parallel to the sides of the square. Then for the equation to the
circle take x*+y* = 2a*, and for the equation to the conic take
y* — a* = \(x*—a a ). The equation to the tangent to the circle at
the point (m l9 y x ) is xx k + yy x = 2a 2 . The equation to the tangent
to the conic at the point (a/, y'Jis yt/ — Xxx* = a* (l — X). These
equations must represent the same line. Hence eliminating A
and x k and y x we shall arrive at an equation which determines
the required locus. It will be found that this equation may be
written
{(«» + y" _ 2a*)\ {a 9 (at* + </*) - 2*V} = 0.
56. The former part follows from Art. 288* For the latter
part proceed thus. Let a perpendicular be drawn from H on
324 ABSWERS TO THE EXAMPLES.
the tangent TQ, and let R denote the intersection, of this per-
pendicular with SQ produced. Then Sll=SQ + QE = 2a; and
TR = TH. We have to £ud the value of the perpendicular from
T on SR ; denote it by )■ ; then r2a - twice the area of the trian-
gle TSR. Let TS=e v and TR or TS=e t ; then bj using the
known expression for the area of a triangle in terms of its sides,
we have 4ra = J(%c'c*+8a'c'+&a'c l s -c l l -c l t - 16a*). This will
lead to the required result. Or thus. Let i/i denote the angle
between HP and TP; then we shall have r = TP sin = TP * -^.
where CD is conjugate to CP ; see Arts. 181 and 193.
may be shewn by Art. 208 that
\VDJ a ' + b'
CHAPTER XV.
And it I
6- s/o+^+Vy^O. 10. Tho equation to
tion being lfiy+mya. + ?ia{j=Q, that to A'Jiia (m + h) a + ly = 0,
that to A'C is (m + n) a + l/l = 0, and that to A'ff
(m + n)a + (l + n){i~ny = Q. 13. fow*V|j
21. r- + — + - = 0. 24. Suppose the focus S is to lie on tin 1
X fj. v
line la + m/3 + «y = 0. Let a', (X, y" denote the vs
respectively for the other focus M of one of the ellipses. Then, by
Art. 181, aa - ^ — yy — the square of the semi-axis minor. Henoe,
substituting in the given equation we obtain -? + -^ -t- — = 0, thai
is, lf¥y' + irhy'a.'+na.'f}' = 0. This shews that the locus of U is
conic section passing through the angular points of the triangle.
25. It will be found that the conio sections may be reprt^
sented by the eqtiations
(1) ft-a'.O, (2) r «-/i" = 0, (3) a/3-/=0.
Now, (I) may be written j3 (y + - 2a) - (o - p)'= 0,
(2) y(« + y-2«-(/S-7>'-°.
W atfJ + .-2 y )-(y-a)'.0
ANSWERS TO THE EXAMPLES. 325
his shews that the tangents to the conio sections at the common
loint are given by
y + /?-2a = 0, a + y-2£ = 0, /? + a-2y = 0;
bese three lines intersect respectively the lines a = 0, fi = 0, y = 0,
a three points which all lie in the line a + fi + y = 0. Again, (1) may
ie written /? (y + 4a + 4/?) -(a + 2)8)* = 0, and (2) may be written
(y+4a + 4/?)-()8 + 2a)* = 0; and this shews that y + 4a + 4/?=0
i a common tangent of (1) and (2), and this common tangent
leets y = at the point where ft + a — 2y = meets it. And so on.
26. The equation to the first hyperbola is /?y = AA '* sin* -^- ;
milarly for the others. 27. See Art. 274.
28 and 29. These may be solved by taking oblique axes coin-
■ding with the sides of the triangle. For instance, consider 29.
7e have aa + bfi + cy = — ab sin G. Thus the equation may be
written cnafi - (J/J + ma) (ab sin G + aa + ty8) = 0; and taking CA
>r the axis of x, and (72? for the axis of y, we have a = a; sin C f ,
? = y sin (7. Substitute for a and /? and then to the equation in
and y we may apply the ordinary test ; see Arts. 272 and 278.
30. £ + £ + ^ = 0.
31. u (mri — m'n) = v (nl' — n'l) = w (lm'— I'm).
32. Let S 1 = be the equation to the inscribed circle, S a =
he equation to the circumscribed circle, these equations not being
Lecessarily in their simplest forms; see Art. 110. Then, if h be
, suitable constant, S t — kJ3 a = will represent the line required.
!n this way we shall have
A B G B G
a* cos 4 — + /2* cos 4 -o + y* cos 4 ^ - 2/fy cos 9 -~ cos 2 -~
- 2ya cos* -^ cos* g- - 2a/J cos* ^ cos* ~
- k (fiy sin -4 + ya sin 2? + aft sin (7)
= (aa + 6)8 + cy) (Ja + wi)8 + ny),
where I, m, n are to be found. Then by comparing like terms we
can find l } m, n.
326
AN8WERS TO THE EXAMPLES.
33. It may be shewn that the equation
nP+my __ly + na
represents a diameter ; for this equation represents a line passing
through the intersection of the tangents at A and B, and through
the middle point of AB. Hence the centre of the conic section is
determined by
nf$+my __h + na _ ma+l/3
a ~~ b " c '
and then the required equation can be found. It is
P = y
m (al — bm + en) n(al+ bm — en) *
34. Assume for the required equation y = constant, that ii
y =k (aa + bfi + cy). Then by applying the result of Example 21 i*
shall obtain for the required equation (lb + ma) (aa + o/3) — nafry =&
35. The equation to the conic section may be taken to be
aft = ky*; and the equation to the line PQ will be a — fi = 0. U*
equation to the chord will be a — fi = hfy. Thus k (a — fif = IPefi
will represent the lines joining P with the points of intersection of
the chord and the conic section. From the symmetrical form of
the last equation we infer that one line makes the same angle
with the line a = which the other makes with the line £ = 0.
N ,'
THE END.
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,::, ,!.'
ill the capacity of a
'.,-. nl.il
Essayist. The Irani
;..'..i ,',,;.
nil melodious., elegant
nil :-, I.„-
The Greek Tenth, well
the note
are clear and useful.
— GvAH-
By C. MERTVALE, B
Sallust.
With English Notes
Second
Edition. 172 pp.
Fcap. 8vo, is. Bd.
(1868).
"Thi*
School edition of Sallu
t iipre-
isely wh
it 11,,. -si-huul edition ii
oght lo be. No nsrirss word.
m it, and no words Ural ,-..nld
1 ::. ■ i
,;' ..'.",.-
-lj-.-ii.!:.
collated with the best
■ILU.O,-..
I Trinity well-selected s
12
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