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  4. If the roots of the equation (a/x-a)+(b/x-b)=1 are equal in magnitude and opposite in sign, then

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Q. If the roots of the equation $\frac{a}{x-a}+\frac{b}{x-b}=1$ are equal in magnitude and opposite in sign, then

Complex Numbers and Quadratic Equations

Solution:

Given equation i s $\frac{a}{x-a}+\frac{b}{x-b}=1$
$\Rightarrow \, a\left(x-b\right)+b\left(x-a\right)$
$=\left(x-a\right)\left(x-b\right)$
$\Rightarrow \, x^{2}-x\left(a+b\right)+ab$
$=ax-ab+bx-ab$
$\Rightarrow \, x^{2}-2x\left(a+b\right)+3ab=0$
So, sum of roots $=\alpha+\left(-\alpha\right)=2\left(a+b\right)$
or $a+b=0$.