Question

If $a{x}^2+2bx+c=0,a\ne 0$ and $d{x}^2+2ex+f=0,d\ne 0$ have a common root and $a,b,c$ are in G.P., then $\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is A A.P.

Given : $a,b,c$ are in G.P., so $b^2=ac$
Now,
$a{x}^2+2bx+c=0$
Checking the discriminant, we get
$D=4b^2-4ac=0(\because b^2=ac)$
This quadratic equation has real and equal roots, let it be $\alpha$
Sum of roots
$2\alpha =-\frac{2b}{a}\Rightarrow \alpha =-\frac{b}{a}$

Thus $d{x}^2+2ex+f=0$ has one root as $\alpha$
$d{\alpha }^2+2e\alpha +f=0\Rightarrow \frac{d{b}^2}{a^2}-2\frac{eb}{a}+f=0\Rightarrow d{b}^2+f{a}^2=2eab$
Dividing by $a{b}^2$, we get
$\Rightarrow \frac{d}{a}+\frac{fa}{b^2}=\frac{2e}{b}\therefore \frac{d}{a}+\frac{f}{c}=\frac{2e}{b}(\because b^2=ac)$

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