First slide

Geometric progression

Question

If a, b, c are positive real numbers that are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $a / d, b / e, c / f$ are in

Moderate

A

A.P.
B

G.P.
C

H.P.
D

none of these

Solution

As $b^{2} = a c, a x^{2} + 2 b x + c = 0$ can be written as
$a x^{2} + 2 \sqrt{a c} x + c = 0 \Rightarrow (\sqrt{a} x + \sqrt{c})^{2} = 0$
$\Rightarrow x = - \sqrt{\frac{c}{a}}, - \sqrt{\frac{c}{a}}$
$∴ - \sqrt{\frac{c}{a}}$ is a root of $d x^{2} + 2 e x + f = 0$
$\Rightarrow d (\frac{c}{a}) - 2 e \sqrt{\frac{c}{a}} + \int = 0$
$\Rightarrow \frac{d}{a} - 2 e \sqrt{\frac{1}{a c}} + \frac{f}{c} = 0$
$\Rightarrow 2 \frac{e}{b} = \frac{d}{a} + \frac{f}{c} \Rightarrow \frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.

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