Question

If $a,b,c$ are in GP, then the equation $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, if $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

• A. AP
• B. GP
• C. HP
• D. A.G.P.

Answer 1

The correct option is A AP

We have, $a{x}^2+2bx+c=0$
$\Rightarrow a{x}^2+2\sqrt{ac}x+c=0(\because b^2=ac)$
$\Rightarrow {(x\sqrt{a}+\sqrt{c})}^2=0$

$\Rightarrow x=-\frac{\sqrt{c}}{\sqrt{a}}$
This satisfies $d{x}^2+2ex+f=0$
$\Rightarrow d\left[\frac{c}{a}\right]+2e[-\frac{\sqrt{c}}{\sqrt{a}}]+f=0$
$\Rightarrow (\frac{dc}{a}+f)=2e\sqrt{\frac{c}{a}}$
$\Rightarrow (\frac{d}{a}+\frac{f}{c})=2e\sqrt{\frac{1}{ac}}$
$\Rightarrow \frac{d}{a}+\frac{f}{c}=\frac{2e}{b}$
Therefore, $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ arein AP.