Question

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

• A. GP
• B. HP
• C. AGP
• D. AP

Answer 1

Answer: (B)

HP

Let roots of the given equation are $\alpha$ and $\beta$
Given that
$\frac{1}{(\alpha {)}^2}+\frac{1}{(\beta {)}^2}=\alpha +\beta$
$\Rightarrow \frac{(\alpha +\beta {)}^2-2\alpha \beta }{\left(\alpha {)}^2\right(\beta {)}^2}=\alpha +\beta$
$\alpha +\beta =-\frac{b}{a}\to \left(i\right)$
$\alpha \beta =\frac{c}{a}\to \left(ii\right)$
Using equations $\left(i\right)$ and $\left(ii\right)$, we get

$\frac{\frac{b^2}{a^2}-\frac{2c}{a}}{\frac{c^2}{a^2}}=\frac{-b}{a}$

$a{b}^2+b{c}^2=2a^{2}c$
$\frac{2a}{b}=\frac{c}{a}+\frac{b}{c}$
Therefore, $\frac{a}{c},\frac{b}{a},\frac{c}{b}$ are in H.P
Hence, option 'B' is correct.