Question

If three distinct numbers $a,b,c$ are in G.P. and the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then which one of the following statements is correct ?

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

Answer 1

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

$a,b,c$ are in G.P.
Let $a=A,b=Ar,c=A{r}^2$
$a{x}^2+2bx+c=0$
$\Rightarrow A{x}^2+2Arx+A{r}^2=0$
$\Rightarrow A(x+r{)}^2=0$
$\Rightarrow x=-r$

$x=-r$ is also a root of $d{x}^2+2ex+f=0$
$\therefore d{r}^2-2er+f=0$
$\Rightarrow d\left(\frac{c}{a}\right)-2e\left(\frac{c}{b}\right)+f=0$
$\Rightarrow \frac{d}{a}-\frac{2e}{b}+\frac{f}{c}=0$
$\Rightarrow \frac{d}{a}+\frac{f}{c}=\frac{2e}{b}$
$\Rightarrow \frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.