41. An ellipse is inscribed in a triangle A, B, C; prove that if a, b, c be the points of contact, the straight lines Aa, Bb, Cc will pass through the same point.

42. Two ellipses, whose major axes are equal, have a common focus; prove that they intersect in two points only.

43. In an ellipse Pp, Dd are conjugate diameters; E is taken in Pp so that PE: Ep=CD2 : CP2; EF is drawn parallel to Dd, meeting the normal PF; GFH being any chord of the ellipse, prove that GPH is a right angle.

44. A parallelogram is inscribed in an ellipse, and from any point on the ellipse two straight lines are drawn parallel to the sides of the parallelogram; prove that the rectangles under the segments of these straight lines, made by the sides of the parallelogram, will be to one another in a constant ratio.

45. Normals at P and D, the extremities of semi-conjugate diameters meet in K; show that KC is perpendicular to PD.

46. The tangent at a point P of an ellipse meets the auxiliary circle in a point Q', to which corresponds Q on the ellipse. Prove that the tangent at Q cuts the auxiliary circle in the point corresponding to P.

47. The locus of a point which cuts parallel chords of a circle in a given ratio, is an ellipse having double contact with the circle.

48. YSZ is drawn through a fixed point S, meeting two fixed straight lines in Y, Z. Prove that the envelope of the circle on YZ is an ellipse having S for focus.

49. If a chord parallel to the axis meet the ellipse in P, P', and if P'Q, P'Q' be chords equally inclined to the axis, then QQ is parallel to the tangent at P.

50. PSP, QCq are any two parallel chords through the focus and centre of an ellipse, prove that

SP.Sp: CQ.Cq= CB2 : CA2.

51. If the diameter conjugate to CP meet SP, HP in E, F; then SE=HF and the circles described about SCE, HCF are equal.

52. The common diameters of two equal, similar, and concentric ellipses are at right angles to one another.

53. If CM, MP be the abscissa and ordinate of any point P on a circle whose centre is C, and if MQ be taken equal to MP and inclined to it at a constant angle, the locus of Q is an ellipse.

54. The tangent at a point P of an ellipse meets the tangents at the vertices in V, V'; on VV' as diameter a circle is described, which intersects the ellipse in Q, Q'; show that the ordinate of Q is to the ordinate of P as CB to CB+CD.

55. From the extremity P of the diameter PQ, in an ellipse, the tangent TPT" is drawn meeting two conjugate diameters in T, T". From P, Q the lines PR, QR are drawn parallel to the same conjugate diameters. Prove that the triangle PQR is to CA.CB as CA.CB to the triangle CTT".

56. PCP' is any diameter of an ellipse. The tangents at any two points D and E intersect in F. PE, P'D intersect in G. Show that FG is parallel to the diameter conjugate to PCP'.

57. SQ, HQ are drawn perpendicular to a pair of conjugate diameters. The locus of Q is a concentric ellipse.

58. A parabola of given latus rectum is described touching symmetrically two conjugate diameters of an ellipse; find the locus of the focus.

59. If AQ be drawn from one of the vertices of an ellipse perpendicular to the tangent at any point P, prove that the locus of the intersection of PS, QA will be a circle.

60. TP, TQ are tangents to an ellipse at the points P, Q. Prove that SP, HP, SQ, HQ are tangents to a circle described with Tas centre.

61. Supplemental chords PL, PL' are equally inclined to a chord PQ, normal at P. Prove that LL' bisects PQ.

62. A, B, C are three points in a straight line; with A, B as foci an ellipse is, described passing through C, and with B and C as foci another ellipse is described passing through A and intersecting the former in P. If PN be drawn perpendicular to CA, prove that AP+CP=PN+CA.

63. If the normal at P in an ellipse meet the axis minor in G, and if the tangent at P meet the tangent at the vertex A in V; show that

SG: SC-PV: VA.

64. ABC is an isosceles triangle of which the side AB is equal to the side AC. BD, BE, drawn on opposite sides of BC and equally inclined to it, meet CA in D, E. If an ellipse is described round BDE having its axis minor parallel to CB, then AB will be a tangent to the ellipse.

65. Show that, if the distance between the foci of an ellipse be greater than the length of its axis minor, there will be four positions of the tangent for which the area of the triangle included between it and the straight lines drawn from the centre of the curve to the feet of the focal perpendiculars upon the tangent, will be the greatest possible.

66. Prove that the distance between the two points on the circumference of an ellipse at which a given chord, not passing through the centre, subtends the greatest and least angles, is equal to the diameter which bisects the chord.

67. In Ex. 61, prove that LP+ PL' is constant.

68. The rectangle contained by the radii of the inscribed and circumscribing circles of the triangle SPH varies as the square of the conjugate diameter.

69. The ordinates of all points on an ellipse being produced in the same ratio, determine the locus of their extremities.

70. The central radii of an ellipse being produced in a constant ratio, the locus of their extremities is an ellipse.

H

CHAPTER VI.

THE HYPERBOLA.

The definition on p. 1 applies to the hyperbola, the ratio spoken of being in this case a ratio of greater inequality.

Let the curve cut the axis in A, A'. Bisect AA' in C. Take a point H in the axis such that CH=CS, where S is the given focus. Then, for a reason which will appear (Prop. IV.), H is called a focus.

Thus S, H are the Foci. Also C is the Centre, and A, A' are the Vertices.

It has been shown, on p. 5, that a straight line drawn parallel to the axis of a hyperbola meets the curve in two points which are situated on opposite sides of the directrix, so that the hyperbola consists of two branches having their convexities in opposite directions.

Compare the first figure on p. 10.

It is hence evident that no straight line drawn perpendicular to the axis and intersecting it between the vertices will meet the curve.

It will however appear that, in the case of the hyperbola, the points B, B', determined as follows, correspond to the extremities of the minor axis in the ellipse.

Through C draw a straight line perpendicular to the axis, and on it take points B, B', equidistant from C, such that CB2 CS-CA2.

=

Then AA' is called the Major and BB' the Minor

Axis.