56. From any point Toutside an ellipse two tangents TP and TQ are drawn to the ellipse; shew that a circle can be described with T as centre so as to touch SP, HP, SQ, HQ, or these lines produced.

If x and y are the co-ordinates of T, shew that the radius of the circle is

√(a3y2+b2x2 — a2b3)

a

T. C. S.

18

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.

29

Let the axis of x 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,, h,, respectively, and those of the second two points k, k, respectively; also let h, k be the co-ordinates of the remaining point. Suppose (Art. 269)

.........

ax2 + bxy + cy3 + 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

From (4) and (5) we find

1

c =

k,k,'

e = - k12 + k2.

Since no

then from (6) we can determine the value of b. three of the five given points are in the same straight line, none of the quantities h, h, k, k2, 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

lvw+mwu+nuv = 0........

.......

·(1),

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 occur in 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=0 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; suppose h, k, 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 h2 and k2, we have two equations from which we can find the values of 7 and 7; suppose 7=p and 27= Substitute these values in (1), which becomes

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

m

(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), (h2, k2). 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);

we will now determine where (1) meets the straight line represented by

By combining (2) with (1) we deduce nuv = 0; therefore either u=0, or v=0; but by taking either of these suppositions and making use of (2), we see that the other supposition must also hold; hence the line (2) meets the curve (1) in only one point, namely, the point of intersection of u=0 and v = 0.

Hence (2) is the tangent to (1) at this point. Similarly mw+nv=0 is the tangent to (1) at the point of intersection of w=0 and v = 0, and nu + lw=0 is the tangent at the point of intersection of u 0 and w =

=

0.

305. The demonstration of the preceding article is imperfect, because we know from Arts. 132, 222, that a line parallel to the axis of a parabola or to either asymptote of an hyperbola meets the curve in only one point, but is not the tangent at that point. The proposition may however be established in the following manner. Take the axis of a coincident with the line u=0, so that u becomes qy, where q is some constant; also take the axis of y coincident with the line v=0, so that v becomes px, where p is some constant. Suppose w = Ax+By+ C. Then (1) of the preceding article be

comes

(Ax+ By + C) (lpx + mqy) + npqxy = 0.

By Art. 283 the equation to the tangent at the origin, that is, at the intersection of x=0 and y = 0, is lpx + mqy = 0, or lv+mu=0; which was to be proved.

306. Let each of the three tangents in Art. 304 be produced to meet the opposite side of the triangle formed by the lines u = · 0, v = 0, w = 0; then it may be shewn that the three points of intersection lie on the straight line

The lines joining the angular points of the triangle formed by the tangents with the angular points of the original