If a, b, c are in geometric progression and the roots of the equations ax² +2bx+c = 0 are $\alpha$ and $\beta$ and those of cx² +2bx+a = 0 are $\gamma$ and $\delta$ then

$\because b^2=ac$ the roots of both the equations are equal.
$\therefore \mathrm{\alpha }=\mathrm{\beta }$ and $\mathrm{\gamma }=\mathrm{\delta }$ but  $\gamma =\frac{1}{\alpha }:\delta =\frac{1}{\beta }$as the given 
equations are reciprocal to each other $\therefore \gamma \delta =\frac{a}{c}$ then $\mathrm{c\gamma }=\mathrm{a\beta }\mathrm{a\alpha }=\mathrm{a\beta }=\mathrm{c\gamma }=\mathrm{c\delta }$