If $\frac{x^2-bx}{ax-c} = \frac{k-1}{k+1}$ has roots, whose magnitude is equal but signs are opposite.

Question

If $\frac{x^2-bx}{ax-c} = \frac{k-1}{k+1}$ has roots, whose magnitude is equal but signs are opposite.

Answer is $\frac{a-b}{a+b}$

I used cross multiplication and since the roots are opposite in sign, on adding the roots, the total must be zero. But this is a long method. Please tell me any shorter method to solve this problem.

Answer 1

Assuming we need the value of $k,$

$$(k+1)x^2-x[b(k+1)-a(k-1)]+c(k-1)=0$$

So, if $\alpha$ is a root, $-\alpha$ will be the other

$$\implies \alpha+(-\alpha)=\dfrac{b(k+1)-a(k-1)}{k+1}$$

$$\implies\dfrac{k-1}{k+1}= \dfrac ab$$

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