Question

If $\alpha ,\beta$ be the non-zero roots of $a{x}^2+bx+c=0$ and ${\alpha }^2,{\beta }^2$ be the roots of $a^2{x}^2+b^{2}x+c^2=0$, then $a,b,c$ are in

• A. A.P
• B. G.P
• C. H.P
• D. None of the above.

Answer 1

The correct option is B G.P

${ax}^2+bx+c=0hasroots\alpha ,\beta \alpha +\beta =-\frac{b}{a}\alpha \beta =\frac{c}{a}{a}^2{x}^2+b^{2}x+c^2=0hasroots{\alpha }^2,{\beta }^2{\alpha }^2+{\beta }^2=\frac{-b^2}{a^2}{\alpha }^2{\beta }^2=\frac{c^2}{a^2}{\alpha }^2+{\beta }^2={(\alpha +\beta )}^2-2\alpha \beta \frac{-b^2}{a^2}=\frac{b^2}{a^2}-\frac{2c}{a}\frac{2c}{a}=\frac{{2b}^2}{a^2}c=\frac{b^2}{a}ac=b^2\therefore a,b,cinG.P$