Question

If a, b, c are in G.P. then the equations $ax^{2} + 2bx + c = 0$ and $dx^{2} + 2ex + f = 0$ 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 of these

Answer 1

Answer: (A)

A.P.

Answer 2

As given, $b^{2} = ac$ ⇒ $ax^{2} + 2bx + c = 0$ can be written as $ax^{2} + 2\sqrt{ac}x + c = 0$ ⇒ $(\sqrt{a}x + \sqrt{c})^{2} = 0$ ⇒ $x = - \sqrt{\frac{c}{a}}$

This must be common root by hypothesis

So it must satisfy the equation, $dx^{2} + 2ex + f = 0$

⇒ $d\left( \frac{c}{a} \right) - 2e\sqrt{\frac{c}{a}} + f = 0$

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

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