Question

If the roots of the equation $\frac{\alpha }{x-\alpha }+\frac{\beta }{x-\beta }=1$ be equal in magnitude but opposite in sign, then $\alpha +\beta$ is equal to:

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is A 0

Given equation $\frac{\alpha }{x-\alpha }+\frac{\beta }{x-\beta }=1$ can be rewritten as $x^2-2(\alpha +\beta )x+2\alpha \beta =0$
Let roots be ${\alpha }^{\prime }$ and $-{\alpha }^{\prime }$
$\Rightarrow {\alpha }^{\prime }+(-{\alpha }^{\prime })=2(\alpha +\beta )$
$\Rightarrow 0=2(\alpha +\beta )$
$\Rightarrow \alpha +\beta =0$