Question

If a, b, c are in G.P, then the equations ax² + 2bx + c = 0 and dx² + 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

a, b, c are in G. P ⇒ b² = ac.

Now the equation ax² + 2bx + c = 0 can be rewritten as

ax² + 2$\sqrt{ac}$x + c = 0 ⇒ ($\sqrt{a}$x +$\sqrt{c}$)² = 0 ⇒ x = –$\sqrt{\frac{c}{a}}$.

If the two given equations have a common root, then this root must be

– $\sqrt{\frac{c}{a}}$. Thus d$\frac{c}{a} - 2e\sqrt{\frac{c}{a}} + f = 0$

⇒ $\frac{d}{a} + \frac{f}{c} = \frac{2e}{c\sqrt{\frac{c}{a}} = \frac{2e}{\sqrt{ac}} = \frac{2e}{b}}$⇒$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A. P