Question

If the cubic equation $x^3+p x^2+q x+r=0$ where $p, q, r \in R$ has root $a^2, b^2, c^2$ satisfying $a^2+b^2=c^2$, then the value of $\frac{p^3+8 r}{p q}$ is equal to $\lambda$. Find the value of $\lambda^5$.

Answer 1

$a^2+b^2+c^2=-p $ ...(1)
$a^2 b^2+b^2 c^2+c^2 a^2=q $....(2)
$a^2 b^2 c^2=-r$ ...(3)
also given $a^2+b^2=c^2$ ...(4)
(1) and (4) $\Rightarrow c^2=-\frac{p}{2}$
put $x =-\frac{ p }{2}$ in the cubic to be $t$ the answer