Question

If $a,b,c$ 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 $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

The correct option is A A.P

Let $a,b,c$ be in G.P.

$\Rightarrow b^2=ac$

Also, ${ax}^2+2bx+c=0$

${\Rightarrow ax}^2+2\sqrt{ac}x+c=0$

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

$\Rightarrow \left(\sqrt{a}\right)x+\sqrt{c}=0$

$\Rightarrow x=-\frac{\sqrt{c}}{\sqrt{a}}$
$\therefore -\sqrt{\frac{c}{a}}$ is a root of ${dx}^2+2ex+f=0$
$\Rightarrow d\left(\frac{c}{a}\right)-2e\sqrt{\frac{c}{a}}+f=0$
$\Rightarrow \frac{d}{a}-2e\sqrt{\frac{1}{ac}}+\frac{f}{c}=0$
$\Rightarrow \frac{2e}{b}=\frac{d}{a}+\frac{f}{c}$
$\Rightarrow \frac{d}{a},\frac{f}{c},\frac{e}{b}$ are in A.P.