Question

Let $p$ and $q$ be two non-zero real numbers and $\alpha, \beta$ are two numbers such that $\alpha^3+\beta^3=-p, \alpha \beta=q$, then the quadratic equation whose roots are $\alpha^2 / \beta$ and $\beta^2 / \alpha$ is

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

Answer 1

Answer: (B)

$\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha}=\frac{\alpha^3+\beta^3}{\alpha \beta}=\frac{-p}{q}$ and $\left(\frac{\alpha^2}{\beta}\right)\left(\frac{\beta^2}{\alpha}\right)=\alpha \beta=q$
Thus, required equation is
$x^2-\left(-\frac{p}{q}\right) x+q=0 $
or $q x^2+p x+q^2=0$