If the harmonic mean between a and b be H , then the value of 1 H-a...

If the harmonic mean between

a

and

b

H

, then the value of

\frac{1}{\mathrm{H}-a}+\frac{1}{\mathrm{H}-b}

Options

A $a+b$
B $a b$
C $\frac{1}{a}+\frac{1}{b}$
D $\frac{1}{a}-\frac{1}{b}$

Correct answer

C. $\frac{1}{a}+\frac{1}{b}$

Step-by-step solution

Putting

\mathrm{H}=\frac{2 a b}{a+b}

, we have

\begin{array}{l} \frac{1}{\mathrm{H}-a}+\frac{1}{\mathrm{H}-b} \\ =\frac{1}{\left(\frac{2 a b}{a+b}-a\right)}+\frac{1}{\left(\frac{2 a b}{a+b}-b\right)}=\frac{a+b}{a b-a^{2}}+\frac{a+b}{a b-b^{2}} \end{array}

=\left(\frac{a+b}{b-a}\right)\left(\frac{1}{a}-\frac{1}{b}\right)=\left(\frac{a+b}{b-a}\right)\left(\frac{b-a}{a b}\right)=\frac{a+b}{a b}=\frac{1}{a}+\frac{1}{b}

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