If the sum of the roots of the quadratic equation  $a{x}^2+bx+c=0$  is equal to the sum of squares of reciprocals, then

The given equation is $a{x}^2+bx+c=0$

$\alpha ,\beta$  are the roots of given equation

Sum of the roots : $\alpha +\beta =\frac{-b}{a}$

Product of the roots : $\alpha \beta =\frac{c}{a}$

By the given condition, roots are equal to the sum of squares of reciprocals

 $\Rightarrow \alpha +\beta$ $=\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

$\Rightarrow \alpha +\beta =\frac{{\left(\alpha +\beta \right)}^2-2\alpha \beta }{{\left(\alpha \beta \right)}^2}$

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

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

$\Rightarrow 2a^{2}c=b{c}^2+a{b}^2$

This condition is satisfies Arithmetic progression

So  $a{b}^2,c{a}^2,b{c}^2$  are in A.P. is correct option.