Question

If a, b, c are in GP, 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. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A AP

Since, a, b, c are in GP
$\Rightarrow b^2=ac$
Given, $a{x}^2+2bx+c=0$
$\Rightarrow a{x}^2+2\sqrt{ac}x+c=0$
$\Rightarrow (\sqrt{a}x+\sqrt{c}{)}^2=0\Rightarrow x=-\sqrt{\frac{c}{a}}$
Since, $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have common root
$\therefore x=-\sqrt{\frac{c}{a}}$ must satisfy
$d{x}^2+2ex+f=0$
$\Rightarrow d.\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0$
$\Rightarrow \frac{d}{a}-\frac{2e}{\sqrt{ac}}+\frac{f}{c}=0$
$\Rightarrow \frac{2e}{b}=\frac{d}{a}+\frac{f}{c}$ $[\because b^2=ac]$
Hence, $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in an AP