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

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In numerical examples, it is convenient to put either k± or k2 equal to unity. If the lines ux = Q and w2 = 0 are parallel, the equation k1u1Jrk2u2 = 0 represents all the lines parallel to ux = 0 since they all have the gradient —ajb^ In numerical examples it suffices to choose Ui+k = 0 to represent all the straight lines parallel to I*! = 0 . The straight line w3 = a3x+b3y+c3 = 0 is a member of the pencil, if it can be expressed in the form k^+k^ = 0. kjkz) ux - (k2/k3) u2. 32 ANALYTICAL GEOMETRY Thus three straight lines wx = 0, u2 = 0 and w3 = 0, no two of which are parallel, are concurrent if constants kl9 k2 and kz can be found such that k1u1-\-k2u2-\-kzu3 = 0.

That is, the points Q^a^ ft) and ö2(a2> ße both lie on the straight line xix+yiy+g(x+xd+f{y+yu+c = 0, and so this is the equation of the required chord of contact. 58 ANALYTICAL GEOMETRY Note carefully that this equation has the same form as the equation of the tangent at the point (xl9 y±). However, in the case of this section, the point (xl9 yx) does not lie on the circle. FIG. 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.

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).

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