Question

If a, b, c are in G.P. and the equations $a{x}^2+2bx+c=0=0$ and $d{x}^2+2ex+f=0$ have a common root. Then?

• A. d, e, f are in G.P.
• B. d, e, f are in A.P.
• C. $\frac{a}{d},\frac{b}{e},\frac{c}{f}$ are in A.P.
• D. $\frac{a}{d},\frac{b}{e},\frac{c}{f}$ are in H.P.

Answer 1

The correct option is D $\frac{a}{d},\frac{b}{e},\frac{c}{f}$ are in H.P.

$b^2=ac$
roots of $a{x}^2+2bx+c=0$ are equal i.e., $-\frac{b}{a}$
$d{(-\frac{b}{a})}^2+2e(-\frac{b}{a})+f=0$
$d{b}^2-2bea+f{a}^2=0$
$dc-2eb+fa=0$
divide by ac
$\frac{dc}{ac}-\frac{2eb}{b^2}+\frac{fa}{ac}=0$
$\Rightarrow \frac{d}{a}-\frac{2eb}{b^2}+\frac{fa}{ac}=0$
$\Rightarrow \frac{d}{a}-\frac{2e}{b}+\frac{f}{c}=0$
$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.