Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-px+q=0,$ then find the equation whose roots are $\frac{q}{p-\alpha }$ and $\frac{q}{p-\beta }$

Answer 1

Let $\frac{q}{p-\alpha }=x$

$\Rightarrow \alpha =p-\frac{q}{x}$

So, we replace $x$ by $p-\frac{q}{x}$ in the equation, we get

${(p-\frac{q}{x})}^2-p(p-\frac{q}{x})+q=0$

$\Rightarrow p^2+\frac{q^2}{x^2}-\frac{2pq}{x}-p^2+\frac{pq}{x}+q=0$

$\Rightarrow \frac{q}{x^2}-\frac{p}{x}+1=0$

$\Rightarrow x^2-px+q=0$ is the required equation whose roots are $\frac{q}{p-\alpha }$ and $\frac{q}{p-\beta }$