If A is the arithmetic mean and $G_{1}, G_{2}$ be two geometric means between any two numbers, then $\frac{G_{1}^{2}}{G_{2}} + \frac{G_{2}^{2}}{G_{1}}$ is equal to :

2 A

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A

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3 A

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None of these

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Solution

The correct option is A $2 A$
$\begin{matrix} l e t t h e n o . b e x, y x, A, y \Rightarrow A = \frac{x + y}{2} a n d, g e o m e t r i c m e a n s i n s e r t e d : x, G_{1}, G_{2}, y \Rightarrow {G_{1}}^{2} = G_{2} x, {G_{2}}^{2} = G_{1} y \Rightarrow \frac{{G_{2}}^{2}}{G_{2}} + \frac{{G_{1}}^{2}}{G_{1}} = x + y ∴ x + y = 2 A \Rightarrow \frac{{G_{2}}^{2}}{G_{2}} + \frac{{G_{1}}^{2}}{G_{1}} = 2 A s o t h a t t h e c o r r e c t o p t i o n i s A . \end{matrix}$

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