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

Answer 1

Answer: (B)

Given ${\alpha }^3+{\beta }^3=-pand\alpha \beta =q$
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
$x^2-\left(\frac{-p}{q}\right)x+q=0$
$\Rightarrow x^2+\frac{p}{q}x+q=0\Rightarrow q{x}^2+px+q^2=0$