By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

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**Example text**

The straight line y = mx+5\/(l +m2) always touches the given circle. This line passes through (—2, 11) if 11 = -2m+5V(l+m2) That is, (2m+ll) 2 = 25(l+m2) which simplifies to 21m 2 -44w-96 = (3/n+4) (7m-24) = 0. 4 24 Hence m = — ^ or 7 and so the tangents are given by 4 x + 3 j - 2 5 = 0 and 24x-7>>+125 = 0 respectively. EXAMPLES 19. Obtain the points of intersection of the straight line 3x—y+5 = 0 and the circle x2+y2- 25 = 0. 20. Obtain the equations of the tangents to the circle x2+y2 = 10 which are parallel to the line y—3x = 7.

Thus the necessary and sufficient condition that ΑλΑ2 touch the circle is τ12 = x^+y^+gixi+xù+Ayi+yù+c = o. r, y) of all points on the tangent at Ax satisfy the equation 7\ = xix+yiy+g(x+xù+f(y+yi)+c = o. That is, the equation of the tangent at Αλ to the circle S = 0 has the equation 7\ = 0. It is worthy of note that the equation 7\ = 0 is obtained from S = 0 by changing x2 into x±x, y2 into y±y9 2x into x+Xi and 2y into y+yv Second Method: The centre C of the circle is at (—g, —f) and so the gradient of CA1 is Oi+ZVCxi+g).

The proof of the result of this section is not valid when Ρ±Ρ2 is parallel to u = 0. Devise a proof to fit this particular case. 17. Perpendicular distance of a point from a straight line We wish to calculate the perpendicular distance d = PN (Fig. 14) from the point /'(xiJi) to the straight line ax+by+c = 0. The equation of the straight line PN is b(x-xj) - a{y-yù = 0. Let the coordinates of N be (α, β). Then A(a-*i) - a(ß-yu Since N lies on ax+by+c can be written = 0. = 0, we have aa+bß+c <*(<>>—xd+Wß—yu = = 0 which —tßXi+byi+c).