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

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Comparison of co efficients yields 44 = a ; l1m2-\-l2m1 = 2h ; m1m2 = b ; m^+m^ nxn2 = c ; = 2f ; 4«2+4«i = 2g. By calculation, we obtain %fgh = (4^2+4^ι) (m^+mfa) = 2 (4«2+4"i) 2 2lJ2m1m2n1n2+l1l2(m1 n2 +m22n12) +mMV+/22"i2)+¥2(W+4V) = 2abc+a(4f2-2bc)+b(4g2-2ca)+c(4h2-2ab)9 and so* Δ = abc+2fgh-af2-bg2-ch2 * In determinantal form Δ^ a h g h b f = 0. 8 f c 40 ANALYTICAL GEOMETRY Consider the equation x2+y2 = 0. In this case we have a=b=\, c=f=g=h=0 and so Δ = 0 . Despite this, x2+y2 = 0 does not represent two straight lines.

This quadratic equation then represents the pair of tangents AXLA and AXL2 to the circle. 61 CIRCLE EXAMPLES 43. Show that the pair of tangents from (—1, 3) to the circle x2+y2 = 5 are mutually perpendicular. 44. y2-j-8A-+6y+21 = o and calculate the angle between them. 45. Find the equation of the tangents from (2, —3) to the circle JC 2 +^ 2 + 6 J C - 4 ^ + 8 = 0. 46. Prove that the tangents from the origin to the circle x2 +j>2-\-2gx+2fy + c = 0 are mutually perpendicular if g2 +f2 = 2c. 35. Parametric treatment of the circle Consider the point P on the circle of radius a, centre the origin, and let the angle which OP (Fig.

This quadratic equation gives two values of x corresponding to the two points of intersection of the straight line and the circle. The straight line is a tangent if the two points coincide. In this case the quadratic equation has equal roots. The required con dition is m 2 c 2 -(l+m 2 ) {c2-r2) = 0, which reduces to c2 = r 2 (l+m 2 ). ) It follows that the straight line y= mx+r\/(l+m2) touches the circle x2+y2 = r2 for all values of m. For any value of m there are two such tangents corresponding to the two values of the square root of (1+m2).