Question

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+px+p^3=0(p\ne 0)$. If $(\alpha ,\beta )$ is a point on the parabola $y^2=x$, then roots of the quadratic equation are

• A. $4$ and $-2$
• B. $-4$ and $-2$
• C. $4$ and $2$
• D. $-4$ and $2$

Answer 1

Answer: (A)

$4$ and $-2$

$\alpha ,\beta$ are the roots of $x^2+px+p^3=0$
$(\alpha +\beta )=-p$ ...(i)
$\left(\alpha \beta \right)=p^3$ ...(ii)
Since the point $(\alpha ,\beta )$ lies on the parabola $y^2=x$, therefore ${\beta }^2=\alpha$ ...(iii)
Solving equations (ii) and (iii), we get ${\beta }^3=p^3\Rightarrow \beta =p$ and solving equation (i) and (iii), we get $p^2=-2p\Rightarrow p=-2$.
Therefore, $\alpha =4$ & $\beta =-2$.