PROP. XVII. If CV be the abscissa of any point Q on the hyperbola, measured along the diameter PP', then

and

QV2 : CV2 –· CP2 = CD2 ; CP2,

QV2: PV.VP' = CD3: CP3,

where CD is the conjugate semi-diameter.

The proof of Prop. XVI., p. 109, is applicable. The letters only require to be changed.

COR. Let VQ meet the conjugate hyperbola in Q'. Then Q'V is equal to the abscissa and CV to the ordinate of Q', the corresponding semi-diameters being CD and CP.

PROP. XVIII. If Q be one extremity and V the middle point of a chord, which meets the asymptotes in R,r and is parallel to the semi-diameter CD, then

CD2 = RV2 - QV2 = RQ.Qr.

By the last proposition, alternando,

QV2 : CD2 = CV2 – CP2 : CP“.

Componendo QV2 + CD" : CD = CV2

by similar triangles CVR, CPL.

: CP

But PL = CD", since, by Prop. v., Cor., PL=CD.
Therefore RV2=QV2+CD".

Hence

CD2 = RV" - QV=RQ.Qr. [Euc. II., 5, Cor. Similarly it may be shown, by means of Prop. XVII., Cor., that

CD" = Q'V2 - RV* = RQ'. Q'r,

Q' being the point in which VQ meets the conjugate hyperbola.

K

PROP. XIX. If TPP' be any diameter and TQ the tangent at a point Q, whose abscissa is CV, then

TC.TV-TP.TP'.

When the diameter meets the hyperbola, the method of Prop. x., p. 48, is applicable.

Otherwise, if Prop. XIII., (the proof of which is general) be assumed, then

The following propositions may be proved by the methods applied to the ellipse.

[p. 84-88.

PROP. XX. The rectangle contained by the segments of a chord QR which passes through a fixed point O bears a constant ratio to the square on the parallel semi-diameter CD.

Also the rectangles contained by the segments of any two intersecting chords are to one another as the squares of the parallel semi-diameters.

PROP. XXI. If a circle and a hyperbola intersect in four points their common chords will be equally inclined to the axis of the hyperbola.

PROP. XXII. If the tangent at P meet any diameter in T and the conjugate diameter in t, then

where CD is the conjugate semi-diameter.

PROP. XXIII. If a chord pass through a fixed point the tangents at its extremities will intersect on a fixed straight line.

PROP. XXIV. If from any point t, tpp' be drawn to meet the hyperbola in p, p' and the chord of contact of tangents through t in o, then tpop' will be cut harmonically.

EXAMPLES.

1. If a directrix and an asymptote of a hyperbola intersect in E, then CE= CA.

2. If the tangent at P meet the directrices in K, K', then PK.PK' = CD2, where CD, CP are conjugate semi-diameters.

3. Any two straight lines drawn parallel to conjugate diameters meet the asymptotes in four points which lie on a circle.

4. If the tangent at P cut an asymptote in T, and SP cut the same asymptote in Q, then SQ= ST.

5. Perpendiculars from the foci upon the asymptotes meet the asymptotes on the circumference of the circle described upon the axis.

6. The tangents at A, A' meet the circle upon SH in the asymptotes.

7. If from the point P in a hyperbola PK be drawn parallel to the transverse axis cutting the asymptotes in I, K, then PK.PI=CA2.

8. If from a point P in the hyperbola PN be drawn parallel to an asymptote to meet the directrix in N, then PN=SP.

9. If from a point P, in the hyperbola, PR be drawn parallel to an asymptote to meet the tangent at the vertex in R, then the difference of SP, PR is equal to half the latus rectum.

10. Prove by means of Examples 2, 8, that the rectangle contained by the focal distances of any point on the hyperbola is equal to the square on the parallel semi-diameter.

11. A hyperbola being defined as the locus of a point whose distance from a fixed point equal to its distance from a fixed straight line, measured parallel to any other given straight line, prove that the second line is parallel to an asymptote of the hyperbola.

12. If PQ be any chord, R the point of contact of the parallel tangent, and PD, RE, QH be drawn parallel to one asymptote to meet the other, then CD.CH=CE".

13. With two conjugate diameters of an ellipse as asymptotes a pair of conjugate hyperbolas are described. Prove that, if one hyperbola touch the ellipse, the other will do so likewise, and that the diameters through the points of contact are conjugate.

14. If any two tangents be drawn to a hyperbola, and their intersections with the asymptotes be joined, the joining lines will be parallel.

15. The tangent to a hyperbola, terminated by the asymptotes, is bisected where it meets the curve. Assuming this, prove that the tangent forms, with the asymptotes, a triangle of constant area.

16. The tangent at P meets one asymptote in T, and TQ, drawn parallel to the other, meets the curve in Q. Prove that, if PQ meet the asymptotes in R, R', then RR' will be trisected in the points P, Q.

17. If CP, CD be conjugate semi-diameters, and through C a line be drawn parallel to either focal distance of P, the perpendicular from D upon this line is equal to half the minor axis.

18. PM, PN are drawn parallel to the asymptotes CM, CN, and an ellipse is constructed having CN, CM for semiconjugate diameters. If CP cut the ellipse in Q, the tangents at Q, P to the ellipse and hyperbola are parallel.

19. Any focal chord of a conic is a third proportional to the transverse axis and the diameter parallel to the chord.

20. The difference of two focal chords, which are parallel to the conjugate diameters of a hyperbola, is constant.

21. If straight lines be taken inversely proportional to focal chords of a conic, which include a right angle, the sum or difference of these lines is constant.

22. A chord of a hyperbola which subtends at the focus an angle equal to that between the asymptotes, always touches a fixed parabola.

23. Given two conjugate diameters of a hyperbola; determine the directions of the axes.

24. The radius of a circle which touches a hyperbola and its asymptotes is equal to the part of the latus rectum intercepted between the curve and the asymptote.

25. A line drawn through one vertex of a hyperbola and terminated by two lines drawn through the other, parallel to the asymptotes, will be bisected where it cuts the curve again.

26. If P be a fixed point on a hyperbola and QQ' an ordinate to CP, the circle QPQ will meet the hyperbola in a fixed point.

27. Tangents are drawn to a hyperbola, and the portion of each tangent intercepted by the asymptotes is divided in a constant ratio; prove that the locus of the point of section is a hyperbola.

28. Given the asymptotes and one point on the curve; find the foci and construct the curve.

29. If a line through the centre of a hyperbola meet in R, T, lines drawn parallel to the asymptotes from any point on the curve, then, the parallelogram PQRT being completed, is a point on the hyperbola.