Question

The cubic equation whose roots are the AM, GM and HM of the roots of $x^2-2px+q^2=0$ is-

• A. $(x-p)(x-q)(x-p-q)=0$
• B. $(x-p)(x-\mid q\mid )(px-q^2)=0$
• C. $x^3-(p+\mid q\mid +\frac{q^2}{p})x^2+(p\mid q\mid +q^2+\frac{\mid q{\mid }^3}{p}x-\mid q{\mid }^3=0$
• D. None of these.

Answer 1

Answer: (B), (C)

The correct options are
B $(x-p)(x-\mid q\mid )(px-q^2)=0$
C $x^3-(p+\mid q\mid +\frac{q^2}{p})x^2+(p\mid q\mid +q^2+\frac{\mid q{\mid }^3}{p}x-\mid q{\mid }^3=0$

Let $\alpha ,\beta$ be the roots of given quadratic equation $x^2-2px+q^2=0$

by the property of the roots we have

$\alpha +\beta =2p,$ and $\alpha \beta =q^2$

AM of the roots are $\frac{\alpha +\beta }{2}$
$\therefore$ AM of roots = p,

GM of roots $=\mid q\mid$,

HM of roots $=\frac{2\alpha \beta }{\alpha +\beta }=\frac{2q^2}{2p}$

$\therefore$ cubic equation is $(x-p)(x+\mid q\mid )(x-\frac{q^2}{p})=0$