Let p and q be two non-zero real numbers and α, β are two numbers such that α3+β3=-p, α β=q, then the quadratic equation whose roots are α2 / β and β2 / α is

Question

Let p and q be two non-zero real numbers and α, β are two numbers such that α3+β3=-p, α β=q, then the quadratic equation whose roots are α2 / β and β2 / α is (A) p x2-q x+p2=0 (B) q x2+p x+q2=0 (C) p x2+q x+p2=0 (D) q x2-p x+q2=0

Tardigrade Admin · Accepted Answer

(α2/β)+(β2/α)=(α3+β3/α β)=(-p/q) and ((α2/β))((β2/α))=α β=q Thus, required equation is x2-(-(p/q)) x+q=0 or q x2+p x+q2=0

Q. Let $p$ and $q$ be two non-zero real numbers and $α, β$ are two numbers such that $α^{3} + β^{3} = - p, α β = q$ , then the quadratic equation whose roots are $α^{2} / β$ and $β^{2} / α$ is

$p x^{2} - q x + p^{2} = 0$

$q x^{2} + p x + q^{2} = 0$

$p x^{2} + q x + p^{2} = 0$

$q x^{2} - p x + q^{2} = 0$

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

Q. Let p and q be two non-zero real numbers and α,β are two numbers such that α3+β3=−p,αβ=q, then the quadratic equation whose roots are α2/β and β2/α is

px2−qx+p2=0

qx2+px+q2=0

px2+qx+p2=0

qx2−px+q2=0