Question

If $p$ and $q$ are non-zero real numbers and $\alpha^{3}+\beta^{3}=-p, \alpha\beta =q, $ then a quadratic equation whose roots are $\frac{\alpha^{2}}{\beta}, \frac{\beta^{2}}{\alpha} is:$

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

Answer 1

Answer: (B)

Given $\alpha^{3}+\beta^{3} =-p \, and \, \alpha\beta =q$
Let $\frac{\alpha^{2}}{\beta} \, and \, \frac{\beta^{2}}{\alpha} $ be the root of required quadratic equation.
So, $\frac{\alpha^{2}}{\beta}+\frac{\beta^{2}}{\alpha}=\frac{\alpha^{3}+\beta^{3}}{\alpha\beta}=\frac{-p}{q}$
and $\frac{\alpha^{2}}{\beta}\times\frac{\beta^{2}}{\alpha}=\alpha\beta=q$
Hence, required quadratic equation is
$x^{2}-\left(\frac{-p}{q}\right) x+q=0$
$\Rightarrow x^{2}+\frac{p}{q} x+q=0 \, \Rightarrow qx^{2}+px+q^{2}=0$