Question

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

• A. AP
• B. HP
• C. GP
• D. None of these

Answer 1

The correct option is A

AP

Explanation for the correct option:

Given that, $a,b,c$ are in GP, then

$b^2=ac$

$\Rightarrow b=\sqrt{ac}$

$a{x}^2+2bx+c=0$

$\Rightarrow$$a{x}^2+2\surd acx+c=0$

$\Rightarrow$ ${(\surd ax+\surd c)}^2=0$

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

$\because a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have common root.

then, $x=-\sqrt{\frac{c}{a}}$ must satisfy $d{x}^2+2ex+f=0$

$\Rightarrow$$d\times \left(\frac{c}{a}\right)–2e\left(\sqrt{\frac{c}{a}}\right)+f=0$

$\Rightarrow$ $\left(\frac{d}{a}\right)–\left(\frac{2e}{\surd ac}\right)+\left(\frac{f}{c}\right)=0$

$\Rightarrow$ $\frac{2e}{b}=\frac{d}{a}+\frac{f}{c}$

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

Hence, Option ‘A’ is Correct.