Assertion -If G be the geometric mean between two distinct positive numbers a and b - then 1-G-a-1-G-b-1-G Reason- Geometric mean G between a b is - ab

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Continuity of a Function

Assertion :If...

Question

Assertion :If $G$ be the geometric mean between two distinct positive numbers $a$ and $b$ , then $\frac{1}{G - a} + \frac{1}{G - b} = \frac{1}{G}$ Reason: Geometric mean $G$ between $a$ & $b$ is $\sqrt{a b}$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Both Assertion and Reason are incorrect

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Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
$L . H . S = \frac{1}{G - a} + \frac{1}{G - b}, w h e r e G = \sqrt{a b}$
$= \frac{1}{\sqrt{a b} - a} + \frac{1}{\sqrt{a b} - b}$
$= \frac{1}{\sqrt{a}} \cdot \frac{1}{(\sqrt{b} - \sqrt{a})} - \frac{1}{\sqrt{b}} \cdot \frac{1}{(\sqrt{b} - \sqrt{a})}$
$= \frac{\sqrt{b} - \sqrt{a}}{\sqrt{a b} (\sqrt{b} - \sqrt{a})} = \frac{1}{\sqrt{a b}} = \frac{1}{G} = R . H . S$
$∴$ Assertion (A) is true followed by Reason (R).

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Assertion :If G be the geometric mean between two distinct positive numbers a and b , then 1G−a+1G−b=1G Reason: Geometric mean G between a & b is √ab

Assertion :If $G$ be the geometric mean between two distinct positive numbers $a$ and $b$ , then $\frac{1}{G - a} + \frac{1}{G - b} = \frac{1}{G}$ Reason: Geometric mean $G$ between $a$ & $b$ is $\sqrt{a b}$