If a, b, c are positive real numbers that are in G.P., then the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$have a common root if $a/d , b/e , c/f$ are in 

As $b^2=ac,a{x}^2+2bx+c=0$ can be written as 

$a{x}^2+2\sqrt{ac}x+c=0\Rightarrow (\sqrt{a}x+\sqrt{c}{)}^2=0$

$\Rightarrow x=-\sqrt{\frac{c}{a}},-\sqrt{\frac{c}{a}}$

$\therefore -\sqrt{\frac{c}{a}}$ is a root of $d{x}^2+2ex+f=0$

$\Rightarrow d\left(\frac{c}{a}\right)-2e\sqrt{\frac{c}{a}}+\int =0$

$\Rightarrow \frac{d}{a}-2e\sqrt{\frac{1}{ac}}+\frac{f}{c}=0$

$\Rightarrow 2\frac{e}{b}=\frac{d}{a}+\frac{f}{c}\Rightarrow \frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.