Explanation

(c)
Let $\alpha$ and $\beta$ are the roots of the given equation
Then, $\alpha + \beta = - \frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$
Also given, $\alpha + \beta = \frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}$
$= \frac{(\alpha + \beta)^{2} - 2\alpha\beta}{\alpha^{2}\beta^{2}}$
$\Rightarrow \ \ \left( - \frac{b}{a} \right) = \left( \frac{- b/a}{c/a} \right)^{2} - \frac{2}{c/a}$
$\Rightarrow \ - \frac{b}{a} = \left( \frac{b}{c} \right)^{2} - \frac{2a}{c}$
$\Rightarrow \ \ \frac{2a}{c} = \frac{b}{c}\left( \frac{b}{c} + \frac{c}{a} \right)$
$\Rightarrow \ \ \frac{2a}{b} = \frac{b}{c} + \frac{c}{a}$
$\Rightarrow \ \ \ \frac{c}{a},\ \frac{a}{b},\ \frac{b}{c}$ are in AP
$\Rightarrow \ \frac{a}{c},\ \frac{b}{a},\ \frac{c}{b}$ are in HP