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

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40. If the chord of contact of the pair of tangents from P to the circle x2-\-y2 = a2 always touches the circle x2-\-y2—lax = 0 show that the locus of P is the curve given by y2 = a(a—2x). 41. Prove that the chord of contact of the tangents to the circle x2+y2+2gx+2fy+c = 0 from the origin and (g,f) are parallel. 42. Obtain the coordinates of the points of contact of the tangents from (2, 0) to the circle Λ: 2 +>' 2 -2Λ:+6>'+5 = 0. 34. Pair of tangents from a point to a circle We saw in section 30 that the two points of intersection Pl9 P2 of A±A2 and the circle x*+y*+2gx+2fy+c=0 are given by the roots of the Joachimsthal quadratic equation S1A12+2r12A1A2+S'2A2a = 0, where the two roots in λ2/λ1 correspond to the two ratios Λ ι Λ / Λ Λ and A&IPzAz.

21 Illustration: Obtain the coordinates of the point of intersection of the tangents to the circle x2+y2—x+y—2 = 0 at the points of intersection with the straight line 5x—3y+l = 0. It is possible to solve for the coordinates of the points of inter section of the circle and the straight line, then to write down the equations of the tangents and to solve these equations. It is more instructive to use the method of this section. Let the required point be (xl9 yj. Then the chord joining the points of contact of the tangents from (xl9 j x ) to the circle is 59 CIRCLE This straight line is identical with 5x-3y+l =0, and so 5 - 3 1 Thus x± and yx satisfy the equations 3^i+5ji = — 1, 3x±— 5yx = - 1 1 whose solution is x1 = —2 and j ^ = 1.

Calculate the ratio ABI AC and explain the significance of the sign of the result. 63. The straight line given by x = t cos ψ—g, y = t sin ψ—/cuts the curve x2+y2+2gx+2fy+c = 0. Determine the values of t at the points of inter section and show that they are independent of ψ. Can you deduce anything about the curve from this result ? 64. Find the equation of the chord of the curve 3x2+4y2 = 28 whose mid point is the point (1, 1). Find also the length of this chord. ) 65. From the point P(l, 3) a line is drawn perpendicular to the line 8JC— 14y—31 = 0 to meet it in Q and PQ is produced to R so that PQ = QR.