33. If from any point in the tangent to a conic section a perpendicular be dropped upon the line joining the focus and the point of contact, prove that the distance of the point in the tangent from the directrix is to the distance of the foot of the perpendicular from the focus as 1 : e.

34. Upon a given straight line as latus rectum, let any number of conic sections be drawn, and from the focus let two straight lines be drawn intersecting them all; then the chords of all the intercepted arcs will, if produced, pass through a single point.

35. The ratio of the sines of the angles made by a diameter of an hyperbola with the asymptotes is equal to the ratio of the sines of the angles made by the conjugate diameter with the same asymptotes respectively.

36. In any conic section if r and be focal distances at right angles to each other, and 7 be half the latus rectum, then

37. Two conic sections equal in every respect are placed with their axes at right angles and with a common focus S; SP, SQ, being radii vectores of the one and the other at right angles to each other, shew that the tangents at P and Q intersect in a straight line passing through the focus and equally inclined to the axes of the curves.

If SPQ were a straight line, then the locus would be a circle.

38. S and H are the foci of an ellipse, and round S, H, as focus and centre, another ellipse is described, having its minor axis equal to the latus rectum of the former. Through any point P in the first draw SPQ to meet the second; it is required to find the locus of the intersection of HP and the ordinate QM.

39. A and B are the centres of two equal circles; AP, BQ, radii of these circles at right angles. ~ If AB2 = 2AP2,

the line PQ always passes through one of the points of intersection of the circles.

40. Tangents are drawn to a conic section at the points P, R; another tangent is drawn at an intermediate point Q, and meets the other tangents in M, N; shew that the angle MSN is half the angle PSR, S being a focus.

41. In a parabola the angle between any two tangents is half the angle subtended at the focus by the chord of contact.

42. A triangle is formed by the intersections of three tangents to a parabola; shew that the circle which circumscribes this triangle passes through the focus.

43. Given a focus and two tangents to a conic section, shew that the chord of contact passes through a fixed point.

44. A circle is described upon the minor axis of an ellipse as diameter; find the locus of the pole with respect to the ellipse of a tangent to the circle.

45. In a parabola two focal chords PSP, QSq, are drawn; shew that a focal chord parallel to PQ will meet pq produced on the tangent at the vertex.

CHAPTER XV.

ABRIDGED NOTATION.

301. Through five points, no three of which are in one straight line, one conic section and only one can be drawn.

Let the axis of a pass through two of the five points, and the axis of y through two of the remaining three points. Let the distances of the first two points from the origin be h,, hy respectively, and those of the second two points k, k2, respectively; also let h, k, be the co-ordinates of the remaining point. Suppose, (Art. 269),

ax2 + bxy + cy2+ dx+ey + 1 = 0............. (1),

to be the equation to a conic section passing through the five points. Since the curve passes through the points (h,, 0), (h,, 0), we have from (1)

Similarly, since the curve passes through (0, k), (0, k2),

we have

Lastly, since the curve passes through (h, k), we have

ah2 + bhk + ck2 + dh + ek + 1 = 0...........................................(6).

From (2) and (3) we find

........

then from (6) we can determine the value of b. Since no three of the five given points are in the same straight line, none of the quantities h, h, k, k, h, k, can be zero; hence the values of the coefficients a, b, c, d, e, are all finite. If we substitute these values in (1), we obtain the equation to a conic section passing through the five given points. As each of the quantities a, b, c, d, e, has only one value, only one conic section can be made to pass through the five given points.

302. The investigation of the preceding article may still be applied when three of the given points are in one straight line; the point (h, k) for instance may be supposed to lie on the line joining (0, k,) and (h,, 0); the conic section in this case cannot be an ellipse, parabola, or hyperbola, since these curves cannot be cut by a straight line in more than two points; the conic section must therefore reduce to two straight lines, namely the line joining the three points already specified, and the line joining the other two points. If, however, four of the given points are in one straight line, the method of the preceding article is inapplicable; it is obvious that more than one pair of straight lines can then be made to pass through the five points.

303. We shall now give some useful forms of the equations to conic sections passing through the angular points of a triangle or touching its sides.

Let u= 0, v=0, w=0, be the equations to three straight lines which meet and form a triangle; the equation

where l, m, n, are constants, will represent a conic section described round the triangle; also by giving suitable values to l, m, n, the above equation may be made to represent any conic section described round the triangle. This we proceed to prove.

I. The equation (1) is of the second degree in the variables x and y, which enter into the expressions u, v, w; hence (1) must represent a conic section.

II. The equation (1) is satisfied by the values of x and y, which make simultaneously v = 0, w = 0; the conic section therefore passes through the intersection of the lines represented by v=0 and w= 0. Similarly the conic section passes through the intersection of w=0 and u= 0, and also through the intersection of u = 30 and v= : 0. Hence the conic section represented by (1) is described round the triangle formed by the intersection of the lines represented by u=0, v=0,

w= 0.

III. By giving suitable values to l, m, n, the equation (1) will represent any conic section described round the triangle. For let S denote a given conic section described round the triangle; take two points on S neither of which is on a side of the given triangle; suppose h1, k1, the co-ordinates of one of these points, and h, k, those of the other. If we first substitute h, and k, for x and y respectively in (1), and then substitute h, and k, we have two equations from which we can find the values of 27 and 27; suppose

m

m

で
=p and

Substitute these values in (1), which becomes

vw+pwu+quv = 0.........

n

(2);

this is therefore the equation to a conic section which has five points in common with S, namely the three angular points of the triangle and the points (h, k), (h, k). The conic section (2) must therefore coincide with S by Art. 301. Hence the assertion is proved.

We might replace one of the constants in (1) by unity, but we retain the more symmetrical form; (1) may be written

304. Equation (1) of the preceding article may be written w (lv +mu) + nuv = 0.................. ....(1);