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

Answer 1

Answer: (C)

$b^2=ac$
Also roots of $a{x}^2=2bx+c=0$ are equal
$\Rightarrow x=\frac{-b}{a},$ common root
$\Rightarrow d{\left(\frac{-b}{a}\right)}^2+2e\left(\frac{-b}{a}\right)+\int =0$
$d{b}^2-2eab+f{a}^2=0,b^2=ac$
$\Rightarrow dac-2eab+f{a}^2=0$
$\Rightarrow dc-7eb+fa=0$
Dividing by ac
$\Rightarrow \frac{d}{a}-\frac{2e}{b}+\frac{f}{c}=0$
$\Rightarrow \frac{d}{a}+\frac{f}{c}2.\frac{e}{b}$