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

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Obtain the lengths of the tangents from the origin to the circle = 0. x2+y2+7x-4y+16 14. Calculate the length of the tangents from (5, 12) to the circle x2 -f y2=69. 50 ANALYTICAL GEOMETRY 28. Circle with given diameter We now obtain the equation of the circle on the line joining the points A(*i, Ji) and A2(x2, y2) as diameter. Let P(x,y) be a variable point (Fig. 18) on this circle. The gradient of AXP is (y—j>i)/(x—x±) whilst the gradient of A2P is (y-y^Kx-xà· Y Χ' 0 Θ X Υ' FIG. 18 Since ΑλΑ2 is a diameter, the angle AXPA2 is a right angle and so the product of the gradients of the perpendicular lines ΑλΡ and ^ 2 P i s - 1 .

7. Find the coordinates of the circumcentre of the triangle formed by the straight lines 3*-j>-5 = 0, x+2j>-4 = 0 and 5jt+3j> + l = 0. 36 ANALYTICAL GEOMETRY 8. Find the equation of the locus of a point equidistant from (xu y±) and 9. Show that the line joining the points (xl9 yd and (x2, y2) will subtend a right angle at (x3,yz) if (*s—*i) (*s—*2> + (ys—yù (y*—yd = 0. Hence, obtain the equation of the circle on the line joining (xl9 yd and (x2, y2) as diameter. 10. Prove that the straight line (A+2)*+(3A-l)j;+A = 0, where λ is a variable, passes through a fixed point and find its coordinates.

16. Prove that the line-pair x2+4xy+y2 = 0 and the straight line x+y = k form an equilateral triangle. 17. Show that the two line-pairs 1 0 J C 2 + 8 ^ + ^ 2 = 0 and 5x2 + 12xy+6y2 = 0 contain the same angle. 24. Bisectors of the line-pair ax 2 +2hxy+by 2 = 0 Let the line-pair ax2+2hxy+by2 = 0 represent the two dis tinct straight lines ^x+m^ = 0 and / 2 x+w 2 j = 0. As in the previous section we have λ/^2 = a ; A(/1/w2+/2Wi) = 2h ; Xm1m2 = b, where λ is some factor of proportionality. The pair of bisectors is given by ΙχΧ+mjy , hx+m^y STRAIGHT LINES 43 That is, (42+^22) (kx+rrny?