38. Any circle described on a focal chord as diameter touches the directrix.

39. If the focus of the parabola be the origin, shew that the equation to the tangent at (x', y') is

yy' = 2a (x + x + 2a).

40. If the focus of a parabola be the origin, shew that the equation to a tangent to the parabola is

41. Two parabolas have a common focus and axis, and a tangent to one intersects a tangent to the other at right angles; find the locus of the point of intersection.

42. If a chord of the parabola y=4ax be a tangent of the parabola y2 = 8a (x - c), shew that the line x=c bisects that chord.

43. From any point there cannot be drawn more than three normals to a parabola.

44. In a parabola whose equation is y=4ax, the ordinates of three points such that the normals pass through the same point are y1, y2 y3; prove that y1+y2+ y1 = 0. Shew also that a circle described through these three points passes through the vertex of the parabola.

29

45. If two of the normals which can be drawn to a parabola through a point are at right angles, the locus of that point is a parabola.

46. If two equal parabolas have the same focus and their axes perpendicular to each other, they enclose a space whose length PQ twice the latus rectum, and breadth

=

latus rectum
√/2

47. Find the length of the perpendicular from an external point (h, k) on the corresponding chord of contact.

48. From an external point (h, k) two tangents are drawn to a parabola; shew that the length of the chord of

contact is

(k2 + 4a2)3 (k2 — 4ah)*

a

49. From an external point (h, k) two tangents are drawn to a parabola; the area of the triangle formed by the tan(k2 — 4ah)

gents and chord is

2a

50. Tangents to a parabola TP, Tp are drawn at the extremities of a focal chord; PG, pg are normals at the

1

1

same points. Shew that + is invariable; and that PG2 pg2

the normals subtend equal angles at T.

51. Two equal parabolas have the same axis, but their vertices do not coincide. If through any point O on the inner curve two chords of the outer curve POp, QOq, be drawn at right angles to one another, then invariable.

1

+

1

PO.Op QO. Oq

is

52. A circle described upon a chord of a parabola as diameter just touches the axis; shew that if 0 be the inclination of the chord to the axis, 4a the latus rectum of the parabola, and c the radius of the circle,

53. If 0, 0' be the inclinations to the axis of the parabola of the two tangents through (h, k), shew that

54. If two tangents be drawn to a parabola so that the sum of the angles which they make with the axis is constant,

the locus of their intersection will be a straight line passing through the focus.

55. Shew that the two tangents through (h, k) are represented by the equation

56.

(k2 — 4ah) (y2 — 4ax)

= {ky — 2a (x + h)}o.

Shew that the lines drawn from the vertex to the

points of contact of the tangents from (h, k) are represented

by the equation

hy2 = =2x (ky―2ax).

CHAPTER IX.

THE ELLIPSE.

158. To find the equation to the ellipse.

The ellipse is the locus of a point which moves so that its distance from a fixed point bears a constant ratio to its distance from a fixed straight line, the ratio being less than unity.

Let S be the fixed point, YY the fixed straight line. Draw SO perpendicular to YY'; take O as the origin, OS as the direction of the axis of x, OY as that of the axis of y.

Let P be a point on the locus; join SP; draw PM parallel to OY and PN parallel to OX. Let OS=p, and let e be the ratio of SP to PN. Let x, y be the co-ordinates of P.

е

By definition,

that is,

SP=e. PN;

:. SP2 = e3PN2;

:. PM2 + SM2 = e3PN3,

y3 + (x − p)2 = e2x2.

This is the equation to the ellipse with the assumed origin and axes.

159. To find where the ellipse meets the axis of x, we put y = 0 in the equation to the ellipse; thus

A and A' are called the vertices of the ellipse, and C, the point midway between A and A', is called the centre of the ellipse.

160. We shall obtain a simpler form of the equation to the ellipse by transferring the origin to A' or C.