Question

If a,b,c are in G.P, 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. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A

A.P.

As given $b^2=ac\Rightarrow equationa{x}^2+2bx+c=0$ can be written as $a{x}^2+2\sqrt{acx}+c=0$

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

This must be the common root by hypothesis.

So it must satisfy the equation $d{x}^2$ + 2ex + f = 0

$\Rightarrow d\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0\Rightarrow \frac{d}{a}+\frac{f}{c}=\frac{2e}{c}\sqrt{\frac{c}{a}}=\frac{2e}{b}\Rightarrow \frac{d}{a},\frac{e}{b},\frac{f}{c}areinA.P$