51. If TP, TQ be tangents to an ellipse at P, Q, then the angles TSP, TSQ are equal, and also the angles THP, THQ. 52. Prove also that the angles STP, HTQ are equal. 53. Tangents at the extremities of two conjugate diameters meet in T; prove that ST, HT intersect the diameters in points which lie on the circumference of a circle. 54. The external angle between any two tangents to an ellipse is equal to the semi-sum of the angles which the chord joining the points of contact subtends at the foci. 55. P, Q are points in two confocal ellipses at which the line joining the foci subtends equal angles; prove that the tangents at P, Q are parallel. 56. A quadrilateral circumscribes an ellipse; prove that either pair of opposite sides subtends supplementary angles at either focus. 57. TP, TQ are tangents to an ellipse, pq any other tangent cutting TP, TQ in p, q; shew that the angle pSq is constant. 58. If two circles touch each other internally, the locus of the centres of circles touching both is an ellipse whose foci are the centres of the given circles. 59. If two ellipses have a common focus, the straight lines joining the points of intersection pass through the intersection of the directrices. 60. A tangent PT to an ellipse meets the tangent at the vertex in T, SQ is drawn perpendicular to ST meeting TA in Q, and QR parallel to ST meeting PT produced in R; shew that R lies on a straight line. 61. If an ellipse be inscribed in a triangle, so that the centre of the circumscribing circle is one focus, the other focus will be the intersection of perpendiculars from the angular points on the opposite sides, and the length of the major axis will be equal to that of the radius of the circumscribing circle. THE HYPERBOLA. 1. Prove that the points of trisection of conterminous circular arcs lie on branches of two hyperbolas. 2. A, A' are the extremities of the axis major of an ellipse, PP' a double ordinate; if AP, A'P' meet, when produced, in Q, Qwill lie on an hyperbola having the same axes as the ellipse. 3. A circle always passes through a fixed point, and cuts a given straight line at a given angle: shew that the locus of its centre is an hyperbola. 4. The locus of the centre of a circle which touches two circles externally is an hyperbola. 5. The locus of the centre of a circle, which touches one circle externally and another internally, is an hyperbola. 6. If a circle be described passing through any point. P of an hyperbola, and the extremities of the transverse axis, and the ordinate at P meet this circle again in Q, Q will be on an hyperbola whose conjugate axis is a third proportional to the conjugate and transverse axes of the original hyperbola. What does this proposition become in the case of the rectangular hyperbola? 7. If a tangent at any point of an hyperbola cut the tangents at the vertices A, A', in T, T', then AT.A'T' = BC2. 8. If on the portion of any tangent intercepted between the tangents at the vertices as diameter a circle be described, it will pass through the foci. 9. The tangents drawn to an hyperbola from any external point subtend equal or supplementary angles at either focus. 10. If a circle be inscribed in the triangle SPH, the locus of its centre will be the tangent at the vertex. 11. If a circle be described touching SP, HP produced, and the transverse axis, the locus of its centre will be an hyperbola. 12. If CP, CD be conjugate semi-diameters, and a straight line be drawn through C parallel to SP, the perpendicular from D on this line will be equal to the semi-conjugate axis. 13. Prove that a circle can be described so as to touch the four straight lines drawn from the foci of an hyperbola to any two points on the same branch of the curve. 14. From a given point draw two tangents to an hyperbola. 15. If the normal to an hyperbola at the point P meet the transverse axis in G, and PN be the ordinate of P, then NG: NO:: BC2: AC2. 16. The perpendiculars from a focus on the asymptotes intersect them in the corresponding directrix. 17. If two hyperbolas have common asymptotes, any chord of the one touching the other will be bisected in the point of contact. 18. Tangents are drawn to an hyperbola, and the portion of each tangent intercepted by the asymptote is divided in a constant ratio; the locus of the points of section is an hyperbola. 19. If any two tangents be drawn to an hyperbola, and their intersections with the asymptotes be joined, the joining lines will be parallel. 20. If A, S be the vertex and focus of an hyperbola, and E, R be the points of intersection of the tangent at A, and the directrix corresponding to S, with an asymptote, then SE is parallel to AR. 21. Given an asymptote, a focus, and a point; construct the hyperbola. 22. Given an asymptote, a directrix, and a point; construct the hyperbola. 23. Given two conjugate diameters, construct the hyperbola. 24. From a point R in one asymptote RE is drawn touching an hyperbola in E, and EV, parallel to an asymptote through E cutting a diameter in T, V, RV is joined cutting the hyperbola in P, p; shew that the diameter parallel to Pp is conjugate to CV. 25. PM, PN are drawn parallel to the asymptotes of an hyperbola from a point P on the curve: an ellipse has CM, CN as conjugate semi-diameters: if CP cut the ellipse in Q, the tangent to the ellipse at Q will be parallel to that to the hyperbola at P. RECTANGULAR HYPERBOLA. 1. In the rectangular hyperbola, a diameter is equal to its conjugate. 2. If a perpendicular from the focus on an asymptote meet the asymptote in R, then SR = CA. 3. The distance of a point on a rectangular hyperbola from the centre is a mean proportional between its focal distances. 4. If PG be the normal at P, then PG = CP. 5. If the tangent at P cut the asymptotes in T, t, then Tt=2PG, where PG is the normal at P. 6. If CY be the perpendicular from the centre on the tangent at P, then PCA, CAY are similar triangles. 7. If P be the middle point of a line AB which cuts off a constant area from the corner of a square, then the locus of P is a rectangular hyperbola. 8. If from the point G, where the normal at P cuts the axis, straight lines be drawn to T, t, the points where the tangent at P cuts the asymptotes, the angle TGt is a right angle. 9. If from any point in a rectangular hyperbola straight lines be drawn to the extremities of a diameter, these lines will make equal angles with the asymptotes. 10. If straight lines be drawn from the extremities of any diameter to any point on the curve, the difference between the angles which they make with the diameter will be equal to the angle which it makes with its conjugate. 11. Find the locus of the centre of a rectangular hyperbola, passing through three given points. 12. A triangle is inscribed in a rectangular hyperbola; prove that the hyperbola passes through the intersection of the perpendiculars from the angles on the sides. 13. Any chord of a rectangular hyperbola subtends at the extremities of any diameter angles which are either equal or supplementary. 14. On opposite sides of any chord of a rectangular hyperbola are described equal segments of circles; shew that the four points, in which the circles to which these segments belong again meet the hyperbola, are the angular points of a parallelogram. 1 MISCELLANEOUS PROBLEMS. 1. In any conic section, if PSP' be a focal chord, is constant. 1 + SP SP 2. A series of conic sections have the same focus S and directrix: find the locus of a point P, such that SP : SB is constant, where SB is the semi-latus rectum of the conic on which Plies. 3. Having given a focus, a tangent, and the excentricity of a conic section, shew that the locus of its centre is a circle. 4. The foci of all parabolic sections which can be cut from a given right cone lie on the surface of another right cone, which has the same axis and vertex as the former. 5. Given a right cone and a point within it; there are but two sections which have this point for focus, and the planes of these sections make equal angles with the straight line joining the given point and the vertex of the cone. 6. From a right cone cut an ellipse of given excentricity. 7. Under what conditions is it possible to cut a rectangular hyperbola from a given cone? |