Question

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

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

Answer 1

The correct option is C H.P.

As given , if $\alpha ,\beta$ be the roots of the quadratic equation, then
$\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{{\alpha }^2{\beta }^2}$
$\Rightarrow -\frac{b}{a}=\frac{\left(\frac{b^2}{a^2}\right)-\left(\frac{2c}{a}\right)}{\left(\frac{c^2}{a^2}\right)}=\frac{b^2-2ac}{c^2}$
$\Rightarrow \frac{2a}{c}=\frac{b^2}{c^2}+\frac{b}{a}=\frac{(a{b}^2+b{c}^2)}{a{c}^2}$
$\Rightarrow 2a^{2}c=a{b}^2+b{c}^2\Rightarrow \frac{2a}{b}=\frac{b}{c}+\frac{c}{a}$
$\Rightarrow \frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in $A.P\Rightarrow \frac{a}{c},\frac{b}{a},\frac{c}{b}$ are in $H.P$.