Question

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

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

Answer: (C)

HP

Given, equation is $a{x}^2+bx+c=0$
If $\alpha$ and $\beta$ be the roots of this equation.
Then, according to question
$\alpha +\beta =\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{{\alpha }^2+{\beta }^2}{{\alpha }^2{\beta }^2}$
$=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{{\alpha }^2{\beta }^2}$
$\Rightarrow \frac{-b}{a}=\frac{b^2-2ac}{c^2}$
$\Rightarrow \frac{-b}{a}=\frac{b^2}{c^2}+\frac{b}{a}=\frac{a{b}^2+b{c}^2}{a{c}^2}$
$\Rightarrow 2a^{2}c=a{b}^2+b{c}^2$
$\Rightarrow \frac{2a}{b}+\frac{b}{c}+\frac{c}{a}$
Here $\frac{c}{a},\frac{a}{b}$ and $\frac{b}{c}$ are in AP.
Therefore, $\frac{c}{a},\frac{b}{a}$ and $\frac{c}{b}$ are in HP.