Question

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

• A. $q^2{x}^2-(p^2-2q)x+1=0$
• B. $x^2-(p^2-2q)x+q^2=0$
• C. $q^2{x}^2+(p^2-2q)x+1=0$
• D. None of these

Answer 1

Answer: (A)

$q^2{x}^2-(p^2-2q)x+1=0$

Here $\alpha +\beta =-p,\alpha \beta =q$ and ${\alpha }^2+\alpha p+q=0$
$\Rightarrow \alpha (\alpha +p)=-q\Rightarrow \alpha +p=-\frac{q}{\alpha }$
Similarly $\beta +p=-\frac{q}{\beta }$.
So, $(\alpha +p{)}^{-2}+(\beta +p{)}^{-2}=\frac{{\alpha }^2+{\beta }^2}{q^2}=\frac{p^2-2q}{q^2}$
and $(\alpha +p{)}^{-2}(\beta +p{)}^{-2}=\frac{{\alpha }^2{\beta }^2}{q^4}=\frac{q^2}{q^4}=\frac{1}{q^2}$