22. If a circle be inscribed in the triangle formed by joining any point on a hyperbola to the foci, the locus of its centre is the tangent at the vertex.

23. If P be any point on a hyperbola whose foci are S, H, and a circle be described touching HP produced, SP, and the transverse axis, the locus of its centre will be a hyperbola.

24. CN being the abscissa of a point P, NQ is drawn parallel to AP and meeting CP in Q. Prove that AQ is parallel to the tangent at P.

25. If tangents at P, Q cut off AR, A'R', AL, A'L' from the tangents at the vertices A, A', then AR.A'R' = AL.A'L'.

26. The curve which trisects the arcs of all segments of a circle described on a given base is a hyperbola whose eccentricity is 2.

27. A chord which subtends a right angle at the vertex meets the axis in a fixed point.

28. Draw a normal to a conic from a given point on the axis minor.

29. P is a fixed point on a conic, and from Q, any point in the ordinate of P produced, QYG is drawn, cutting the polar of Q at right angles in Y and meeting the axis in G. Prove that G is a fixed point, and that QG.GY is equal to the square of the normal at P.

30. The chord of contact of tangents to a central conic through an external point P meets the axis in T; PXG is drawn meeting the axis in G and cutting the chord at right angles in X. Prove that CG.GT= ST.HT.

31. Prove also that, if CM, SY, HZ be perpendiculars upon the chord, then CM.PG=CB and SY.HZ=CM.YG. What do these theorems become when the point P lies on the curve?

32. If a pair of the chords of intersection of a circle and a conic be produced to meet a similarly situated conic, the four points of intersection will lie on another circle; and, if the two conics be similar and concentric, the circles will be concentric.

33. The normals to the circle on AA', in a central conic, at the points where the tangent at P meets it, bisect the focal radii to P.

34. If a circle be described through any point P of a given hyperbola and the extremities of the transverse axis, then the ordinate of P, being produced, meets the circle again on a fixed hyperbola. Also the axes of the first hyperbola and the conjugate axis of the second are proportionals.

35. An ellipse and a hyperbola are described so that the foci of each are at the extremities of the transverse axis of the other; prove that the tangents at their points of intersection meet the conjugate axis in points equidistant from the centre.

36. The points of trisection of a series of conterminous circular arcs lie on branches of two hyperbolas; determine the distance between their centres.

37. Given a focus, a tangent, and one point on a hyperbola; determine the locus of the other focus.

38. If, from a fixed point O, OP be drawn to a given circle, and the angle TPO be constant, the envelope of TP is a conic having O for focus.

39. If from the focus of a conic a line be drawn making a given angle with any tangent, find the locus of the point in which it intersects the tangent.

40. Tangents from any point to a system of confocal conics make equal angles with two fixed lines.

41. The locus of the intersection of tangents to a parabola which cut at a given angle, is a hyperbola.

42. A straight line drawn from the focus of a conic so as to make a constant angle with a chord subtending a constant angle at the focus, meets the chord, in general, upon the circumference of a fixed circle.

43. If an ellipse and hyperbola have the same foci and tangents be drawn to the one to intersect at right angles those drawn to the other, the locus of the points of intersection is a circle.

44. P, P' are points on a hyperbola and its conjugate, S, S' the interior foci of the branches on which P, P' lie. Prove that the difference of SP, S'P' is equal to difference of CA, CB.

45. A, P and B, Q are points taken respectively in two parallel straight lines, A, B being fixed and P, Q variable. Prove that if the rectangle AP.BQ be constant, the line PQ will always touch a fixed ellipse or a fixed hyperbola according as P, Q are on the same or opposite sides of AB.

46. Through a fixed point S a straight line SYY' is drawn to meet fixed parallel straight lines in Y, Y'. Prove that the envelope of the circle on YY' is a hyperbola, S being a focus and the fixed lines directrices.

47. PQ is a chord of an ellipse at right angles to the major axis AA'; PA, QA' are produced to meet in R; show that the locus of R is a hyperbola having the same axes as the ellipse.

48. If a hyperbola be described touching the four sides of a quadrilateral inscribed in a circle and one focus lie on the circle, the other focus will also lie on the circle.

49. If TP, TQ be tangents to an ellipse or hyperbola, S, H being the foci; then

ST2: HT" SP.SQ: HP.HQ.

=

50. A point D is taken on the axis of a hyperbola, whose eccentricity is 2, such that its distance from the focus S is equal to the distance of S from the further vertex A'; P being any point on the curve, A'P meets the latus rectum in K. Prove that DK and SP intersect on a certain fixed circle.

CHAPTER VII.

THE HYPERBOLA CONTINUED.

Take any point E (fig., Prop. II.,) on a fixed straight line drawn through the centre C, and let EN meet the axis at right angles in N. Then the ratio of EN to CN is the same whatever be the position of E on the fixed straight line.

If the ratio of EN to CN be equal to the ratio of the semi-axes CB, CA, the straight line CE is called an Asymptote, for a reason which will appear in Prop. I., Cor. 2.

Make the angle NCM equal to NCE. Then CM is the other asymptote.

It follows from the definition given above that, when N coincides with the vertex A, EN becomes equal to the semiminor axis CB.

In this case CE2 = CB2 + CA2 = CS”.

Two hyperbolas are said to be conjugate when the transverse axis of each is the conjugate axis of the other.

Thus, in fig., Prop. VIII., the hyperbola which has CB, CA for semi-axes is conjugate to that which has CA, CB for semi-axes.

It is evident that any two conjugate hyperbolas have the same asymptotes.

Conjugate Diameters and Supplemental Chords may be defined as on p. 75.

Let PP' (fig., Prop. VIII.) be any diameter of a hyperbola, terminated by the curve. Then the conjugate diameter will not meet the curve, but its extremities are defined as the points D, D', in which it meets the conjugate hyperbola.