If $a,\ b$ and $c$ are in geometric progression and the roots of the equation $ax^{2} + 2bx + c = 0$ are $\alpha$ and $\beta$ and those of $cx^{2} + 2bx + a = 0$ are $\gamma$ and $\delta$, then

Question

Dr. Abubacker · Accepted Answer

(c)
$ax^{2} + 2bx + c = 0$
$\Rightarrow ax^{2} + 2\sqrt{ac}x + c = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \lbrack\because b^{2} = acbrack$
$\Rightarrow \ \ \left( \sqrt{ax} + \sqrt{c} ight)^{2} = 0\  \Rightarrow x = \frac{- \sqrt{c}}{\sqrt{a}},\ \frac{- \sqrt{c}}{\sqrt{a}}$
$\Rightarrow a\alpha = a\beta$
Now, $cx^{2} + 2bx + a = 0$
$\Rightarrow \ \ \ cx^{2} + 2\sqrt{ac}x + a = 0$
$\Rightarrow \left( \sqrt{c}x + \sqrt{a} ight)^{2} = 0$
$\Rightarrow \ \ \ \ x = \frac{- \sqrt{a}}{\sqrt{c}} = \frac{- \sqrt{a}}{\sqrt{c}}\  \Rightarrow c\gamma = c\delta$
$	herefore\ \ \ a\alpha = a\beta = c\gamma = c\delta$

Q. If $a,\ b$ and $c$ are in geometric progression and the roots of the equation $ax^{2} + 2bx + c = 0$ are $\alpha$ and $\beta$ and those of $cx^{2} + 2bx + a = 0$ are $\gamma$ and $\delta$, then

Explanation

Reviewed by Dr. Abubacker

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