The parabola therefore is a limiting form of an ellipse or of an hyperbola, the latus rectum of which is finite, but the major and minor axes are infinite. The centre and the second focus are at infinity.

It is a very instructive exercise for the student to deduce the properties of a parabola from those of an ellipse or hyperbola,

158. Let the focus of a conic be on the directrix.

Take the focus as origin, and let the directrix be the axis of y; then the equation of the conic will be

This equation represents two straight lines which are real if e be greater than unity, coincident if e be equal to unity, and imaginary if e be less than unity.

Hence we must not only consider as conics an ellipse, a parabola, and an hyperbola, but also two real or imaginary straight lines.

It should be noticed that the directrix of a circle is at an infinite distance; also that the foci and directrices of two parallel straight lines are all at infinity.

EXAMPLES ON CHAPTER VII.

1. AOB, COD are two straight lines which bisect one another at right angles; shew that the locus of a point which moves so that PA.PB=PC.PD is a rectangular hyperbola.

2. If a straight line cut an hyperbola in Q, Q' and its asymptotes in R, R', shew that the middle point of QQ' will be the middle point of RR'.

3. A straight line has its extremities on two fixed straight lines and passes through a fixed point; find the locus of the middle point of the line.

4. A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area; find the locus of the middle point of the line.

5. OA, OB are fixed straight lines, P any point, and PM, PN the perpendiculars from P on OA, OB; find the locus of P if the quadrilateral OMPN be of constant area.

6. The distance of any point from the centre of a rectangular hyperbola varies inversely as the perpendicular distance of its polar from the centre.

7. PN is the ordinate of a point P on an hyperbola, PG is the normal meeting the axis in G; if NP be produced to meet the asymptote in Q, prove that QG is at right angles to the asymptote.

8. If e, é be the eccentricities of an hyperbola and of the 1 1 conjugate hyperbola, then will + e'

==

1.

9. The two straight lines joining the points in which any two tangents to an hyperbola meet the asymptotes are parallel to the chord of contact of the tangents and are equidistant from it.

10. Prove that the part of the tangent at any point of an hyperbola intercepted between the point of contact and the transverse axis is a harmonic mean between the lengths of the perpendiculars drawn from the foci on the normal at the same point.

11. If through any point O a line OPQ be drawn parallel to an asymptote of an hyperbola cutting the curve in P and the polar of O in Q, shew that P is the middle point of OQ.

12. A parallelogram is constructed with its sides parallel to the asymptotes of an hyperbola, and one of its diagonals is a chord of the hyperbola; shew that the direction of the other will pass through the centre.

13. A, A' are the vertices of a rectangular hyperbola, and P is any point on the curve; shew that the internal and external bisectors of the angle APA' are parallel to the asymptotes.

14. A, A' are the extremities of a fixed diameter of a circle and P, P' are the extremities of any chord perpendicular to this diameter; shew that the locus of the point of intersection of AP and A'P' is a rectangular hyperbola.

15. Shew that the co-ordinates of the point of intersection of two tangents to an hyperbola referred to its asymptotes as axes are harmonic means between the co-ordinates of the points of contact.

16. From any point of one hyperbola tangents are drawn to another which has the same asymptotes; shew that the chord of contact cuts off a constant area from the asymptotes.

17. The straight lines drawn from any point of an equilateral hyperbola to the extremities of any diameter are equally inclined to the asymptotes.

18. The locus of the middle points of normal chords of the rectangular hyperbola x2-y=a2 is (y2 - x2)3 = 4a2x2y3.

19. Shew that the line x = O is an asymptote of the hyperbola 2xy + 3x2 + 4x = 9.

What is the equation of the other asymptote ?

20. Find the asymptotes of xy-3x-2y= 0. What is the equation of the conjugate hyperbola?

21. Shew that in an hyperbola the ratio of the tangents of half the angles which the radii vectores from the foci to a point on the curve make with the axis, is constant.

22. A circle intersects an hyperbola in four points; prove that the product of the distances of the four points of intersection from one asymptote is equal to the product of their distances from the other.

23. Shew that if a rectangular hyperbola cut a circle in four points the centre of mean position of the four points is midway between the centres of the two curves.

24. If four points be taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two, and if a, ß, y, d be the inclinations

to either asymptote of the straight lines joining these points respectively to the centre; prove that tan a tan ẞ tan γ tan 8=1.

tangents to the circle described on the straight line joining the foci of the hyperbola as diameter; shew that the locus of

y3

their poles with respect to the hyperbola is + =

a1

1

b4 a2 + b2.

26. If two straight lines pass through fixed points, and the bisector of the angle between them is always parallel to a fixed line, prove that the locus of the point of intersection of the lines is a rectangular hyperbola.

27. Shew that pairs of conjugate diameters of an hyperbola are cut in involution by any straight line.

28. The locus of the intersection of two equal circles, which are described on two sides AB, AC of a triangle as chords, is a rectangular hyperbola, whose centre is the middle point of BC, and which passes through A, B, C.

CHAPTER VIII.

POLAR EQUATION OF A CONIC, THE FOCUS BEING THE POLE.

159. To find the polar equation of a conic, the focus being the pole.

Let S be the focus and ZM the directrix of the conic, and let the eccentricity be e.

Draw SZ perpendicular to the directrix, and let SZ be taken for initial line.

Let LSL be the latus rectum, then e. SZ SL = 1

suppose.

=