Question

For the equation $\frac{1}{x+a}-\frac{1}{x+b}=\frac{1}{x+c}$, if the product of roots is zero, then the sum of roots is-

Answer 1

$\frac{1}{x+a}-\frac{1}{x+b}=\frac{1}{x+c}$

$⟹\frac{x+b-x-a}{(x+a)(x+b)}=\frac{1}{x+c}$

$⟹x^2+2ax+ab-bc+ac=0$

If products of roots are $0$

$ab+ac-bc=0$

$⟹a=\frac{bc}{b+c}$

Sum of roots are $=-2a$

So, Sum $=-2\left(\frac{bc}{b+c}\right)$