Explanation

(a)
Let the roots are $\alpha,\ \beta,$ so $\alpha + \beta = \frac{- b}{a}$ and $\alpha\beta = \frac{c}{a}$
Now, $\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{\alpha^{2} + \beta^{2}}{\alpha^{2}\beta^{2}}$
$= \frac{(\alpha + \beta)^{2} - 2\alpha\beta}{(\alpha\beta)^{2}}$
$= \frac{\frac{b^{2}}{a^{2}} - \frac{2c}{a}}{\frac{c^{2}}{a^{2}}} = \frac{b^{2} - 2ac}{c^{2}}$
Also, $\alpha + \beta = \frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}$ [given]
$\Rightarrow \ - \frac{b}{a} = \frac{b^{2} - 2ac}{c^{2}}$
$\Rightarrow \ - bc^{2} = ab^{2} - 2a^{2}c$
$\Rightarrow 2a^{2}c = ab^{2} + bc^{2}$
$\Rightarrow ab^{2},\ ca^{2},\ bc^{2}$
Or $bc^{2},\ ca^{2},\ ab^{2}$ are in AP