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Implicit Differentiation

If A1 ,A2 ,...

Question

If $A_{1}, A_{2}, G_{1}, G_{2}$ and $H_{1}, H_{2}$ are two A.Ms, G.Ms and H.M.s between two numbers, then $\frac{A_{1} + A_{2}}{H_{1} + H_{2}} . \frac{H_{1} H_{2}}{G_{1} G_{2}}$ equals to.

A

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B

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C

1

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D

8

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Solution

The correct option is C 1
$A_{1}, A_{2}, G_{1}, G_{2}$ and $H_{1}, H_{2}$ are two A.Ms, G.Ms and H.M.s between $a$ and $b$ .
$a, A_{1}, A_{2}, b$ are in AP $\Rightarrow A_{1} + A_{2} = a + b$
$a, G_{1}, G_{2}, b$ are in GP $\Rightarrow G_{1} G_{2} = a b$
$a, H_{1}, H_{2}, b$ are in HP $\Rightarrow \frac{1}{H_{1}} + \frac{1}{H_{2}} = \frac{1}{a} + \frac{1}{b}$
$\Rightarrow \frac{H_{1} H_{2}}{H_{1} + H_{2}} = \frac{a b}{a + b}$
$\frac{A_{1} + A_{2}}{H_{1} + H_{2}} . \frac{H_{1} H_{2}}{G_{1} G_{2}}$ $= \frac{a + b}{a + b} \cdot \frac{a b}{a b} = 1$
Hence, option C.

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