Question

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

Answer 1

Answer: (A)

A.P

Since, $a,b,c$ are in $G.P.$

$\Rightarrow b^2=ac$

Given, $a{x}^2+2bx+c=0$

$\Rightarrow a{x}^2+2\sqrt{ac}x+c=0\Rightarrow {(\sqrt{a}x+\sqrt{c})}^2=0$

$\Rightarrow x=-\sqrt{\frac{c}{a}}$

Since, $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have common root.

$\therefore x=-\sqrt{\frac{c}{a}}$ must satisfy $d{x}^2+2ex+f=0$

$\Rightarrow d.\frac{c}{a}-2e\sqrt{\frac{c}{a}}+f=0\Rightarrow \frac{d}{a}-\frac{2e}{\sqrt{ac}}+\frac{f}{c}=0$ $(\because b^2=ac)$

$\Rightarrow \frac{2e}{b}=\frac{d}{a}+\frac{f}{c}$

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