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

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39. Find the chord of contact of the tangents to the circle x2-\-y2 —4x—6y + 3 = 0 from the origin and hence prove that the equation of the tangents is x2 + \2xy+6y2 = 0. ) 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.

17. Obtain in normal form the equations of the bisectors of the angles between x cos a+y sin a = p and x cos ß-\-y sin ß = q. 18. Obtain the equations of the diagonals of the parallelogram formed by the four lines ax+by = 0, ax+by+c = 0, Ix+my = 0 and lx-\-my+n = 0. What is the condition that this parallelogram be a rhombus ? 19. Prove that the area of the triangle formed by the three straight lines y =--■ /w^ + fi, y = m2x+c2 and y = m3x-\-c3 is 1 rfe^a) 2 + (Cs^l + (C1-C2)2] 2 Lw 2 —/w 3 inz—mi m1—m2\ 20.

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. 60 ANALYTICAL GEOMETRY The straight line AXA2 is a tangent if both points Ρλ and 1\ coincide with Lx (Fig. 22). In this case, Joachimsthal's quadratic equation FIG. 22 has equal roots and so Thus the locus of the point A2 as Ax is held fixed is given by SXS-T?