Question

If $a,b,c$ are in G.P., then the equations, $a{x}^2+2bx+c=0$ and $dx{x}^2+2ex+f=0$ 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

Answer 1

Answer: (A)

$a,b,c$ are in G.P. $\Rightarrow b^2=ac$
Discriminant of $a{x}^2+2bx+c=0$ is $4b^2-4ac=4ac-4ac=0$
$\Rightarrow$ roots are coincident and equals $-\frac{2b}{2a}=-\frac{b}{a}$
This must be the common root. Hence $x=-\frac{b}{a}$ must satisfy $dx{x}^2+2ex+f=0$
$\therefore d\frac{b^2}{a^2}-2e\frac{b}{a}+f=0\Rightarrow d{b}^2-2eba+f{a}^2=0\Rightarrow dac-2eba+f{a}^2=0$
dividing by $a^{2}c$,
$\frac{d}{a}-\frac{2eb}{ac}+\frac{f}{c}=0$
$\frac{d}{a}-\frac{2e}{b}+\frac{f}{c}=0\left(b^2=ac\right)$
$\Rightarrow \frac{d}{a},\frac{e}{b},\frac{f}{c}are\; inA.P$