Question

The cubic equation whose roots are the AM, GM and HM of the roots of x² - 2px + q² = 0 is

• (A) (x - p)(x - q)(x - p - q) = 0
• (B) (x - p)(x - |q|)(px - q²) = 0
• (C) $x^3-(p+|q|+\frac{q^2}{p})x^2+(p|q|+q^2+\frac{|q{|}^3}{p})x-|q{|}^3=0$
• (D) None of these

Answer 1

The Correct Option is B, C

Explanation:
$\alpha +\beta =2p,\alpha \beta =q^2$$\therefore \mathrm{AM}$ of roots $=\mathrm{p},\mathrm{GM}$ of roots $=|\mathrm{q}|,\mathrm{HM}$ of roots $=\frac{2\alpha \beta }{\alpha +\beta }=\frac{2{\mathrm{q}}^2}{2\mathrm{p}}$$\therefore$ The cubic equation is $(x-p)(x-|q|)(x-\frac{q^2}{p})=0$