Question

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

• A. $x^2+p^{2}x+p^{2}q=0$
• B. $x^2-q^{2}x+p^{2}q=0$
• C. $x^2+q^{2}x+p^{2}q=0$
• D. $x^2-p^{2}x+p^{2}q=0$

Answer 1

The correct option is D $x^2-p^{2}x+p^{2}q=0$

Since $\alpha$ and $\beta$ are roots of the equation
$x^2+px+q=0$, therefore
$\alpha +\beta =-p$ ...(i)
and $\alpha \beta =q$ ...(ii)
Sum of the roots $={\alpha }^2+\alpha \beta +{\beta }^2+\alpha \beta$
$=(\alpha +\beta {)}^2=p^2$
Product of the roots $=({\alpha }^2+\alpha \beta )({\beta }^2+\alpha \beta )$
$=\alpha \beta (\alpha +\beta {)}^2=q{p}^2$
Required equation will be
$x^2$-(Sum of the roots)$x$ + Product of the roots = $0$
or $x^2-p^{2}x+q{p}^2=0$