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Geometric Mean

If G be the g...

Question

If $G$ be the geometric mean of $x$ and $y,$ where $x, y > 0,$ then the value of $\frac{1}{G^{2} - x^{2}} + \frac{1}{G^{2} - y^{2}}$ is

A

G^{2}

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B

- G^{2}

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C

\frac{1}{G^{2}}

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D

- \frac{1}{G^{2}}

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Solution

The correct option is C $\frac{1}{G^{2}}$
Given, $G = \sqrt{x y}$
$\frac{1}{G^{2} - x^{2}} + \frac{1}{G^{2} - y^{2}} = \frac{1}{x y - x^{2}} + \frac{1}{x y - y^{2}} = \frac{1}{x - y} {- \frac{1}{x} + \frac{1}{y}} = \frac{1}{x y} = \frac{1}{G^{2}}$

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