200 The Parabola.

54. Two equal parabolse, S, S't have coincident axes, which have e
same direction, while the focus F of 8 is the vertex of Sr. Show that i P
he a point on £', the chord of S through P, which passes through F, is .e
minimum chord through P.

(Hid.

55. If ti9 £3, £3, #4 denote the tangents of half the inclinations to the i is
of four concyclic tangents to a parabola, t\tzkt± = 1. (NEUBEHG.) (49£

DEF. — jfowr lines are said to be concyclic when they touch the same circl
The tangent at the point 0 to a parabola is x — y tan <p + a tan2<£ = 0 If

this touch the circle (x — a)2 + (y - £)2 = P2 the perpendicular on it from .e

point aj3 is equal to P. Hence we get

a cos2 <£ - j3 sin $ cos $ + « sin2 0 = P cos <£.
Now, putting

1 - tan2 e l-t*

sin A = cos 20 = . - 5- = - 5,
Y 1 + tan20 1 + <2

we get

* - 2 (J2 - ft) $ + 2 (2o - a) t* - 2 (R + j8) * + 0 = 0.
In this equation the roots are #j, fe, #3, ^- Hence the proposition is pro' i.
56. If a circle osculates a parabola, and if 20 be the inclination of le
tangent at the point of osculation, and 20i, of the other common tangenl
tan 0i = cot3 0. (Ibid.) (50<
57. The diameter of the circle inscribed ia the quadrilateral formec »y
concyclic tangents of a parabola is equal to the sum of the perpendici; rs
from the focus on the tangents. (Ibid
For the equation in t gives
2 tan 0 = 2 (E - £)/«, 2 cot 0 = 2 (22 4- £)/«.
Hence, by addition,
4.B/« = 2 (cot e + tan 6) - 22 cosec 20 = 22 sec 0 ;
.•. 27i! = 2^ sec<£ = sum of perpendiculars.
58. The ordinate of the centre of the circle is the arithmetic mean oi 10
sum of the ordinates of the points of contact on the parabola. (Ibid
59. If JEi, J2o, PS, P4 be the radii of curvature at the points of coi ,ct
with the parabola of concyclic tangents,
2a P = «§ (R$ -i-P2i + JB3* + J2$). (I*»tf.) (50
For PI = 2« sec3 4>i, equation (441).
Hence, a sec <j>i = (a2Pi/2)&, &c.
60. If four circles osculate at the points of contact of concyclic tan gc ;s,
.the other common tangents of these circles and the parabola are concycl .