34. Qis a point on the normal at any point P of an ellipse whose centre is C such that the lines CP, CQ make equal angles with the axis of the ellipse; shew that PQ is proportional to the diameter conjugate to CP. 35. If a pair of tangents to a conic be at right angles to one another, the product of the perpendiculars from the centre and the intersection of the tangents on the chord of contact is constant. 36. Find the locus of the middle points of chords of an ellipse which all pass through a fixed point. 37. If P be any point on an ellipse and any chord PQ cut the diameter conjugate to CP in R, then will PQ. PR be equal to half the square on the diameter parallel to PQ. 38. Find the locus of the middle points of all chords of an ellipse which are of constant length. 39. Tangents at right angles are drawn to an ellipse; find the locus of the middle point of the chord of contact. 40. If three of the sides of a quadrilateral inscribed in an ellipse are parallel respectively to three given straight lines, shew that the fourth side will also be parallel to a fixed straight line. 41. The area of the parallelogram formed by the tangents at the ends of any pair of diameters of an ellipse varies inversely as the area of the parallelogram formed by joining the points of contact. 42. If at the extremities P, Q of any two diameters CP, CQ of an ellipse, two tangents Pp, Qq be drawn cutting each other in T and the diameters produced in P, and 9, then the areas of the triangles TQp, TPq will be equal. 43. From the point O two tangents OP, OQ are drawn to x2 y3 the ellipse + is equal to a2 b2 = 1; shew that the area of the triangle CPQ and the area of the quadrilateral OPCQ is equal to (b3h3 + a3k3 — a2b2)3, C being the centre of the ellipse, and h, k the co-ordinates of 0. 44. TP, TQ are tangents to an ellipse whose centre is C, shew that the area of the quadrilateral CPTQ is ab tan (-); where a, b are the semi-axes of the ellipse, and 4, are the eccentric angles of P and Q. 45. PCP' is a diameter of an ellipse and QCQ' is the corresponding diameter of the auxiliary circle; shew that the area of the parallelogram formed by the tangents at P, P', Q, Q' 8a2b where is the eccentric angle of P. is (a - b) sin 26' 46. A parallelogram circumscribes a circle, and two of the angular points are on fixed straight lines parallel to one another and equidistant from the centre; shew that the other two are on an ellipse of which the circle is the minor auxiliary circle. 47. Two fixed conjugate diameters of an ellipse are met in the points P, Q respectively by two lines OP, OQ which pass through a fixed point O and are parallel to any other pair of conjugate diameters; shew that the locus of the middle point of PQ is a straight line. 48. If from any point O in the plane of an ellipse the perpendiculars OM, ON be drawn on the equal conjugate diameters, the direction OP of the diagonal of the parallelogram MONP will be perpendicular to the polar of 0. 49. Three points A, P, B are taken on an ellipse whose centre is C. Parallels to the tangents at A and B drawn through P meet CB and CA respectively in the points Q and R. Prove that QR is parallel to the tangent at P. 50. Find the locus of the point of intersection of normals at two points on an ellipse which are extremities of conjugate diameters. 51. Normals to an ellipse are drawn at the extremities of a chord parallel to one of the equi-conjugate diameters; prove that they intersect on a diameter perpendicular to the other equi-conjugate. 52. If normals be drawn at the extremities of any focal chord of an ellipse, a line through their intersection parallel to the axis-major will bisect the chord. 53. If a length PQ be taken in the normal at any point P of an ellipse whose centre is C, equal in length to the semidiameter which is conjugate to CP, shew that Q lies on one or other of two circles. 54. Shew that, if & be the angle between the tangents to the ellipse x2 y2 55. TP, TQ are the tangents drawn from an external x2 a2 b2 point (x, y) to the ellipse 56. If two tangents to an ellipse from a point T intersect at an angle 4, shew that ST. HT cos &= CT2 — a2 — b3, where C is the centre of the ellipse and S, H the foci. 57. If the perpendicular from the centre C of an ellipse on the tangent at any point P meet the focal distance SP, produced if necessary, in R; the locus of R will be a circle. 58. If two concentric ellipses be such that the foci of one lie on the other, and if e, e' be their eccentricities, shew that √e2 + e22 – 1 their axes are inclined at an angle cos ee' 12 59. Shew that the angle which a diameter of an ellipse subtends at either end of the axis-major is supplementary to that which the conjugate diameter subtends at the end of the axis-minor. 60. If 0, 0' be the angles subtended by the axis major of an ellipse at the extremities of a pair of conjugate diameters, shew that cot20+ cot°0′ is constant. 61. If the distance between the foci of an ellipse subtend angles 20, 20' at the ends of a pair of conjugate diameters, shew that tan20+ tan' is constant. 62. If λ, X' be the angles which any two conjugate diameters subtend at any fixed point on an ellipse, prove that cot λ+cot X' is constant. 63. Shew that pairs of conjugate diameters of an ellipse are cut in involution by any straight line. 64. A triangle whose sides touch an ellipse and enclose it, is a minimum; shew that each side of the triangle touches at its middle point, and that the triangle formed by joining the points of contact is a maximum. 65. A, B, C, D are four fixed points on an ellipse, and P any other point on the curve; shew that the product of the perpendiculars from P on AB and CD bears a constant ratio to the product of the perpendiculars from P on BC and DA. 66. Find the locus of the point of intersection of two normals to an ellipse which are perpendicular to one another. 67. Find the equation of the locus of the point of intersection of the tangent at one end of a focal chord of an ellipse with the normal at the other end. 68. Two straight lines are drawn parallel to the axis-major of an ellipse at a distance ab √a* - b from it; prove that the part of any tangent intercepted between them is divided by the point of contact into two parts which subtend equal angles at the centre. 69. PG is the normal to an ellipse at P, G being in the major axis, GP is produced outwards to Q so that PQ = GP; shew that the locus of Q is an ellipse whose eccentricity is a2 - b2 a2 + b2, and find the equation of the locus of the intersection of the tangents at P and Q. S. C. S. 10 CHAPTER VII. THE HYPERBOLA. Definition. The Hyperbola is the locus of a point which moves so that its distance from a fixed point, called the focus, bears a constant ratio, which is greater than unity, to its distance from a fixed straight line, called the directrix. 139. To find the equation of an hyperbola. Let S be the focus and ZM the directrix. Divide ZS in A so that SA: AZ= given ratio=e: 1 suppose. Then A is a point on the curve. There will also be a point A' in SZ produced such that Let C be the middle point of AA', and let AA′ = 2a. Then |