Question

If three distinct numbers a,b,c are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ 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)

$\frac{d}{a} , \frac{e}{b} , \frac{f}{c}$ are in $A.P$.

Answer 2

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