Question

lf the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to sum of the squares of their reciprocals, then $b{c}^2,c{a}^2,a{b}^2$ are in

• A. A.P
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A A.P

We have, $\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$
$\Rightarrow {\alpha }^2{\beta }^2(\alpha +\beta )={\alpha }^2+{\beta }^2$
$\Rightarrow {\alpha }^2{\beta }^2(\alpha +\beta )=(\alpha +\beta {)}^2-2\alpha \beta$
$\Rightarrow \frac{c^2}{a^2}(-\frac{b}{a})=\frac{b^2}{a^2}-2\cdot \frac{c}{a}$
$\Rightarrow -b{c}^2=a{b}^2-2c{a}^2$
$\Rightarrow 2c{a}^2=b{c}^2+a{b}^2$
Hence $b{c}^2,c{a}^2,a{b}^2$ are in A.P.