If the roots of the equation $\cfrac { { x }^{ 2 }-bx }{ ax-c } =\cfrac { \lambda -1 }{ \lambda +1 } $ are equal and opposite then the value of $\lambda$ is

If the roots of the equation are equal and opposite, their sum is zero, so rearranging the given equation to ax^2 + (c*lambda - b*lambda - b - c)x - ac = 0 means the coefficient of x must be zero. Setting (c - b)lambda - (b + c) = 0 gives lambda = (b + c) / (c - b). The product of these roots is -p^2, which equals the constant term divided by the leading coefficient, so -p^2 = -ac / a, leading to p^2 = c. Substituting c = p^2 into the lambda formula yields lambda = (p + p^2) / (p^2 - p), which factors to p(1 + p) / p(p - 1) = (a - b) / (a + b).

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