Let PT be the tangent at P, and let Pnr parallel to transverse axis cut CD and the conjugate axis in n and r,

nT, rN will be parallelograms. Then since the angles at F and r are right angles, a circle will pass through F, n, r and g.

Hence

Also

and

PG: Pg :: NG : NC :: BC2 ; AC2:

PF. Pg-AC; . PF. PG=BC".

19. The asymptotes of any hyperbola are parallel to the generating lines of the cone in which a plane through the vertex parallel to the cutting plane cuts the cone.

Let VE be the line in which a plane through V parallel to the cutting plane cuts the plane of the paper.

Make VE equal to AC and draw FEF' through E at right angles to the axis.

Then the part of the perpendicular through E to the plane of the paper between that plane and the cone will be the height of a right-angled triangle which has half the angle between two generating lines at the base, call it EG. Then EGEF. EF": and from equal triangles ACD, VEF and

D

A' CD', VEF', EF = CD, EF' = CD'; .. EG2 = CD. CD′ = BC2 and EG = BC. Hence the generating lines are inclined to VE at the same angle as the asymptotes to CA parallel to it, and being in a plane parallel to that containing the asymptotes, are parallel to them.

EXAMPLES.

1. Or all hyperbolas that can be cut from a given cone, the ratio BC AC is greatest in that cut by a plane parallel to the axis.

:

Give a construction for determining, if possible, the direction of the plane that will cut a rectangular hyperbola from a given cone.

3. NPQ is drawn from any point N of the conjugate axis of a rectangular hyperbola at right angles to it, to cut the auxiliary circle and the hyperbola in P and Q ; prove NP2 + NQ2 = 2AC2,

4. Prove the same for any hyperbola and the ellipse on the

same axes.

5. A circle passes through A, A', the vertices of a rectangular hyperbola: the common chord parallel to AA' is a diameter of the circle.

6. The tangent PT to a circle intersects a fixed diameter in T, and TQ is drawn at right angles to the diameter of a length bearing a constant ratio to TP as P moves on the circle, Q will move on an hyperbola which has the given circle for its auxiliary circle.

7. Through any point P of a circle on AB as diameter PA, BP are drawn and produced to Q and R such that QR is at right angles to BA and bisected by it: prove that as P moves on the curve Q, R will move on a rectangular hyperbola.

8. Draw a tangent to an hyperbola parallel to a given line.

9. From a given point in an hyperbola draw a line such that the intercept between the other point of intersection and an asymptote shall equal a given line. When does the problem become impossible?

10. The tangent at P meets an asymptote in T, and TQ is drawn to the curve parallel to the other asymptote. PQ produced both ways meets the asymptotes in R, R'. RR' is trisected in P, Q.

11. From a given point P in a rectangular hyperbola PM, PN are drawn equally inclined to an asymptote, and when produced meet the curve again in Q, R; prove that QR is a diameter.

12. Find the position and magnitude of the axes of an hyperbola which has a given line for asymptote, touches another line in a given point, and passes through another given point.

13. Draw lines from the centre of an hyperbola to the extremities of any chord: the intercepts of any line parallel to the chord between these lines and the asymptotes will be equal.

14. From CV. CTCP deduce in the rectangular hyperbola, VC. VT=QV2.

15. In the rectangular hyperbola the triangles CVQ, QVT are similar.

16. In the rectangular hyperbola, R is the middle point of a chord Qg and RQ', Rq' are drawn parallel to the tangents at q, Q to meet CQ and Cq: shew that a circle will circumscribe CQRq'.

17. From any point R of an asymptote RN, RM are drawn at right angles to the axes intersecting the hyperbola and its conjugate in P and D. Prove CP, CD are conjugate in the rectangular and general hyperbolas.

18. The tangent at P meets an asymptote in T, TN is drawn at right angles to the transverse axis; prove in the rectangular and general hyperbolas that NP passes through D the extremity of the diameter conjugate to CP.

19. Any two tangents have their points of intersection with the asymptotes joined: the lines so drawn will be parallel.

pro

20. From any point P of an hyperbola PH, PK are drawn each parallel to one asymptote to meet the other: these lines duced if necessary meet any line through the centre in R and T. Complete the parallelogram PRQT, and shew that Q is a point on the curve.

21. Draw a tangent to an hyperbola at a given distance (less than AC) from C.

22. The tangent to an hyperbola meets a pair of conjugate diameters in T, t and the second tangents are drawn to the curve from T and t, they will touch it at the extremity of a diameter.

23. An hyperbola can be drawn through the extremities of any two radii of a circle having the diameters at right angles to the radii as asymptotes.

An hyperbola can be drawn through the extremities of any two semi-diameters of an ellipse having the diameters conjugate to them as asymptotes.

25. If PG be the normal at P, CG = 2CN in the rectangular hyperbola.

26. PG. Pg CD3.

27. If the tangent at P intersects the asymptotes in R, r the circle on Gg as diameter will pass through C, R and r.

28. In the same hyperbola Gg varies inversely as the perpendicular from the centre on the tangent.

29. An hyperbola is cut from a given cone, and a straight line drawn from a point of it parallel to an asymptote; the plane through the vertex of the cone and this line will cut the cone in two straight lines one of which is parallel to the line in the curve. Hence prove the proposition of Art. 11.

30. Apply the method of proof in Art. 15 to shew that if parallel tangents at Q, Q' meet the tangent at P in T, T',

QT : PT :: QT : P'T'.

Prove the following propositions in the rectangular hyperbola:

31. The lines bisecting the angles between CP and the tangent at P are parallel to the asymptotes.

32. The tangent at the point Q intersects a pair of conjugate diameters in T, T": prove that CQ is the tangent to the circle round CTT".