Question

If a,b,c are in G.P., 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. None

Answer 1

The correct option is A

A.P.

$\therefore$ $b^2=ac$ so $a{x}^2+2bx+c=0$ has equal roots

Since $\Delta =4b^2-4ac=0$

and the roots is $\frac{-2b}{2a}=\frac{-b}{a}$ .

$\therefore$ $\frac{d{b}^2}{a^2}-\frac{2eb}{a}+f=0$

or $\frac{dac}{a^2}-\frac{2eb}{a}+f=0$

or $\frac{d}{a}-\frac{2eb}{a}+\frac{f}{c}=0$ or $\frac{d}{a}-\frac{f}{c}=\frac{2e}{b}$

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