If the harmonic mean between a and b be H- then the value of 1- H - a -1- H - bA- a-bB- abC- 1-2-1-hD- I - a -1- b

The correct option is C 1/a + 1/bPutting H=2ab/a+b, we have 1/H a + 1/H b = 1/( 2ab/a+b a ) + 1/( 2ab/a+b b ) = a+b/ab a^2 + a+b/ab b^2 = (a+b/b a )(1/a ...

If the harmon...

Question

If the harmonic mean between a and b be H, then the value of $\frac{1}{H - a} + \frac{1}{H - b}$ is

a+b

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\frac{1}{a} + \frac{1}{b}

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\frac{1}{a} - \frac{1}{b}

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Solution

The correct option is C $\frac{1}{a} + \frac{1}{b}$
Putting $H = \frac{2 a b}{a + b},$ we have $\frac{1}{H - a} + \frac{1}{H - b}$
$= \frac{1}{(\frac{2 a b}{a + b} - a)} + \frac{1}{(\frac{2 a b}{a + b} - b)} = \frac{a + b}{a b - a^{2}} + \frac{a + b}{a b - b^{2}}$
$= (\frac{a + b}{b - a}) (\frac{1}{a} - \frac{1}{b}) = (\frac{a + b}{b - a}) (\frac{b - a}{a b}) = \frac{a + b}{a b} = \frac{1}{a} + \frac{1}{b} .$

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