Question

If $a,b$ and $c$ are in G.P., then the quadratic 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.

$a{x}^2+2bx+c=0$
$\Rightarrow D=4b^2-4ac=0$ (as $a,b,c$ are in G.P.)
$\Rightarrow$ equation has equal roots which are equal to $-\frac{b}{a}$ each.

Thus, it should also be the root of $d{x}^2+2ex+f=0$
$\Rightarrow d{\left(\frac{-b}{a}\right)}^2+2e\left(\frac{-b}{a}\right)+f=0$
$\Rightarrow d\frac{b^2}{a^2}-2\frac{be}{a}+f=0$
$\Rightarrow d\frac{ac}{a^2}-2\frac{be}{a}+f=0(\text{as}{b}^2=ac)$
$\Rightarrow \frac{d}{a}+\frac{f}{c}=2\frac{eb}{ac}=2\frac{e}{b}$