If d, e, f are in G.P and two quadratic equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root then $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

Here $e^2=df$

Now $d{x}^2+2ex+f=0$ [given] $\Rightarrow d{x}^2+2\sqrt{df}x+f=0\Rightarrow x=-\sqrt{\frac{f}{d}}$

Putting in $a{x}^2+2bx+c=0$ we get $a\frac{f}{d}+c=2b\sqrt{\frac{f}{d}}\Rightarrow \frac{a}{d}+\frac{c}{f}=\frac{2b}{e}$

$\therefore \frac{a}{d},\frac{b}{e},\frac{c}{f}$ are in A.P.

$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in H.P