EXAMPLES.

1. The locus of the centre of an equilateral hyperbola described about a given equilateral triangle is the circle inscribed in the triangle.

2. PG is the normal at P; GE a perpendicular on CP; prove that PE=PF, F being the point in which the normal meets the diameter parallel to the tangent at P.

3. The tangent from G to the circle on the axis is equal to PG.

4. In a rectangular hyperbola no pair of tangents can be drawn at right angles to each other.

5. An asymptote of a rectangular hyperbola meets the perpendicular upon it from either focus at a distance from the centre equal to half the axis.

6. The distance of any point from the centre is a geometric mean between its distances from the foci.

7. Straight lines drawn from any point on the curve to the extremities of a diameter are equally inclined to the asymptotes.

8. Q is a point on the conjugate axis of a rectangular hyperbola and QP, drawn parallel to the transverse axis, meets the curve in P; prove that PQ=AQ.

9. The locus of the middle point of a line which cuts off a constant area from the corner of a square is a rectangular hyperbola.

10. In a rectangular hyperbola CY is drawn perpendicular to the tangent at P; prove that the triangles PCA, CAY are similar.

11. The foci of an ellipse are situated at the ends of a diameter of a rectangular hyperbola; show that the tangent and normal to the ellipse, at any point where it meets the hyperbola, are parallel to the axes of the latter.

12. If a right-angled triangle be inscribed in a rectangular hyperbola, prove that the hypotenuse is parallel to the normal to the hyperbola at the right angle.

13. If two rectangular hyperbolas touch one another, their common chords through the point of contact will include a right angle and the remaining common chord will be parallel to the common tangent.

14. If a rectangular hyperbola circumscribe a rightangled triangle, the locus of its centre will be a circle passing through one of the angular points.

15. If 44' be any diameter of a circle, PQ any ordinate to it, then the locus of the intersection of AP, A'Q is a rectangular hyperbola.

16. In a rectangular hyperbola, focal chords parallel to conjugate diameters are equal.

17. LL is any diameter of a rectangular hyperbola, P any point on the curve; prove that the external and internal bisectors of the angle LPL' are parallel to fixed straight lines.

18. Straight lines parallel to conjugate diameters meet the asymptotes in four points which lie on a circle.

19. Assuming Prop. XVIII., p. 129, show how to deduce from Prop. v. that in the rectangular hyperbola

CV.CT-CP2.

20. Ellipses are inscribed in a given parallelogram; show that their foci lie on a rectangular hyperbola.

21. If two concentric rectangular hyperbolas be described, the axes of one being asymptotes of the other, they will intersect at right angles.

22. The portion of the tangent intercepted by the asymptotes subtends a right angle at the foot of the normal.

23. If, between a rectangular hyperbola and its asymptotes, any number of concentric elliptic quadrants be inscribed the rectangle contained by their axes will be constant.

24. The base of a triangle ABC remaining fixed, the vertex C moves along an equilateral hyperbola which passes through A and B. If P, Q be the points in which AC, BC meet the circle on AB as diameter, the intersection of AQ, BP is always situated on the hyperbola.

25. Any conic which passes through the four points of intersection of two rectangular hyperbolas, must be itself a rectangular hyperbola.

26. If two concentric rectangular hyperbolas have a common tangent, the lines joining their points of intersection to their respective points of contact with the common tangent, will subtend equal angles at their common centre.

27. If lines be drawn from any point of a rectangular hyperbola to the extremities of a given diameter, the difference between the angles which they make with the diameter will be equal to the angle which it makes with its conjugate.

28. From fixed points A, B straight lines are drawn intersecting in C, such that the difference of the angles CBA, CAB is constant; find the locus of C.

29. Prove that in a rectangular hyperbola the triangle formed by the tangent at any point and its intercepts on the axes, is similar to the triangle formed by the straight line joining that point with the centre, and the abscissa and semiordinate of the point.

30. On opposite sides of any chord of a rectangular hyperbola are described equal segments of circles; show that the four points, in which the circles to which the segments belong again meet the hyperbola, are the angular points of a parallelogram.

31. If a conic be described through the centres of the inscribed and exscribed circles of any triangle, its centre will lie on the circle which circumscribes the triangle.

32. The locus of the centre of a rectangular hyperbola described about a triangle is the circle passing through the middle points of the sides of the triangle.

33. If PQR be a triangle inscribed in a rectangular hyperbola, the intersections of pairs of tangents at P, Q, R lie on the lines joining the feet of the perpendiculars from the angular points of the triangle upon the opposite sides.

34. Given a triangle such that any vertex is the pole of the opposite side with respect to an equilateral hyperbola ; the circle circumscribing the triangle passes through the centre of the curve.

35. A circle, described through the centre of a rectangular hyperbola and any two points, will also pass through the intersection of lines drawn through each of these points parallel to the polar of the other.

CHAPTER IX.

CORRESPONDING POINTS.

Any fixed straight line being taken as axis, if the ordinate NP of a variable point P to be produced, in a constant ratio, to p, then the points p, P correspond, and the locus of either point corresponds to the locus of the other. [fig., Prop. III. Hence, if any other ordinate MQ be produced to q, so that MQ: Mq=NP: Np,

then the points Q, q correspond.

PROP. I. Straight lines correspond to straight lines. Let P, p be corresponding points and let the locus of P be a straight line which meets the axis in T. Join Tp and draw any ordinate MQq, meeting the straight lines TP, Tp in Q, q respectively.

Then

MQ: Mq=NP: Np,

or the points Q, q correspond.

Hence, to any point Q on TP corresponds a point q on Tp. In other words, the straight line Tp corresponds to TP. COR. Corresponding straight lines intersect on the axis.

PROP. II. Tangents correspond to tangents.

Let P, Q be adjacent points on any curve and p, q the corresponding points. Then the straight line pq corresponds to PQ.