5. Prove that, if the conic (la)* + (mß)2 + (ny)* = 0 be a parabola, its focus and directrix are given by the equations

Hence prove that, if a parabola touch three straight lines, its directrix always passes through a fixed point. State, in geometrical language, the position of this point relatively to the three straight lines.

6. A system of parabolas is described so that a given triangle is self-conjugate with respect to each curve of the system; prove that the locus of the focus is a circle, that the directrix always passes through the centre of the circle described about the triangle, and that every parabola of the system touches the three straight lines which bisect each pair of sides of the triangle.

7. If P be any point on the circumference of a circle, O any fixed point, prove that the locus of the point, in which the tangent at P intersects the line which bisects OP at right angles, is a straight line.

8. A rectangular hyperbola circumscribes a triangle; shew that the loci of the poles of its sides are three straight lines forming another triangle, whose angular points lie on the sides of the first, where they are met by perpendiculars from the opposite angular points.

9. If ABC, A'B'C' be two triangles, each of which is selfconjugate with regard to the same given conic, shew that another conic can be described about both.

10. If α, B, Y, d be the distances of a point from four given straight lines, so connected that la + mß+ny + pd = 0, prove that, if a conic be described, touching these four straight lines, the locus of either of its foci will be the curve of the third degree represented by the equation

11. Prove that the polar reciprocal of a rectangular hyperbola with respect to any point S, is a conic, the sum of the squares on the semi-axes of which is equal to the square on the distance of its centre from S.

12. Two given conics are so related that each of their common tangents subtends a right angle at a given point. Prove that, if any two points be taken, one on each conic, so that the line joining them also subtends a right angle at that point, the envelope of this line will be a conic, of which that point is a focus.

13. In Example 2, p. 116, prove that if any conic (4) be drawn touching the directrices of the four conics, the polar of the given point with respect to it will be a tangent to a conic, having the given point as focus and touching the sides of the triangle; and that the tangents from the given point to A are at right angles to each other.

14. If, through a fixed point O, a straight line be drawn cutting the sides AB, AC of a triangle ABC in P, Q respectively, and BQ, CP be joined, prove that the locus of their point of intersection is a conic circumscribing the triangle ABC.

15. If Pɑ, P, Pc be the semi-diameters of a conic, respectively parallel to the sides of the triangle of reference, prove that the area of the conic is

16. PQ is the chord of a conic, having its pole on the chord AB or AB produced; Qq is drawn parallel to AB meeting the conic in q; shew that Pq bisects the chord AB.

17. Similar circular arcs are described on the sides of a triangle ABC, their convexities being towards the interior of the triangle; shew that the locus of the radical centre of the three circles is the rectangular hyperbola

18. Prove that, if r be either semi-axis of the curve represented by the equation

ua2 + vß3 + wy3 + 2u′ By + 2v ́ya + 2w'aß = 0,

the values of r will be the roots of the equation

α

a2

A

b

ca

{u+ " (au_bv_cw)}r" -as cos ▲ {v+ (bv-cw-au')}r-bs cos B

bc

19. If a triangle is self-conjugate with respect to each of a series of parabolas, the lines joining the middle points of its sides will be tangents: all the directrices will pass through 0, the centre of the circumscribing circle: and the focal chords, which are the polars of Q, will envelope an ellipse inscribed in the given triangle which has the nine point circle for its auxiliary circle.

20. A conic circumscribes a triangle ABC, the tangents at the angular points meeting the opposite sides on the straight line DEF. The lines joining any point P on DEF to A, B and C meet the conic again in A', B', C': shew that the triangle ABC envelopes a fixed conic inscribed in ABC, and having double contact with the given conic at the points where they are met by DEF. Also the tangents at A, B, C' to the original conic meet BC, C'A', A'B' in points lying on DEF.

21. The anharmonic ratio of the pencil formed by joining a point on one of two conics to their four points of intersection is equal to the anharmonic range formed on a tangent to the other by their four common tangents.

22. The four common tangents to two conics intersect two and two on the sides of their common conjugate triad.

23. The locus of the centres of conics inscribed in a triangle and such that the centres of the escribed circles form a self-conjugate triad with respect to them is a straight line parallel to aa+bB+cy = 0 in areal co-ordinates.

24. A triangle ABC, right-angled at A, is inscribed in a rectangular hyperbola; tangents at B and C meet in P: prove that AB, AP, AC and the tangent at A form a harmonic pencil.

25. AB, CD are two fixed chords of a conic. A straight line APQ meets CD in P and the curve in Q, and on CQ a point R is taken so that PR subtends a constant angle at B: the locus of R will be a conic passing through B and C.

26. Conics circumscribing a triangle have a common tangent at the vertex; through this point a straight line is drawn: shew that the tangents at the various points where it cuts the curves all intersect on the base.

27. One conic touches OA, OB in A and B, and a second conic touches OB, OC in B and C: prove that the other common tangents to the two conics intersect on AC.

28. With any one of four given points as centre, a conic is described, self-conjugate with regard to the other three; prove that its asymptotes are parallel to the axes of the two parabolas which pass through the four given points.

29. A rectangular hyperbola passes through the angular points, and a parabola touches the sides, of a given triangle; shew that the tangents drawn to the parabola, from one of the points where the hyperbola cuts the directrix of the parabola, are parallel to the asymptotes of the hyperbola. Which of the two points on the directrix is to be taken? When they coincide, shew that either curve is the polar reciprocal of the other with respect to the coincident points.

30. The triangular coordinates of the two circular points at infinity are given by the equations

31. If each of two conics be reciprocated with respect to the other, prove that the four points of intersection of any two of the conics thus obtained, and the four points of intersection of the other two, lie on a conic.

32. With any one of four given points as centre, a conic is described, self-conjugate with regard to the other three; prove that its asymptotes are parallel to the axes of the two parabolas which pass through the four given points.

33. With each of four given straight lines as directrix, two conics are described, self-conjugate with regard to the other three; prove that the four pairs of conics. thus obtained, will have the same pair of points as foci corresponding to the given directrix.

34. If a triangle be self-conjugate to a rectangular hyperbola, and any conic be described, touching the sides of the triangle, each focus of this conic will lie on the polar of the other with respect to the rectangular hyperbola.

CAMBRIDGE: PRINTED AT THE UNIVERSITY PRESS.