Question

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

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

Answer 1

Answer: (D)

$H= \frac{2ab}{a+b}$
$ \frac{1}{H-a}+\frac{1}{H-b}= \frac{1}{\frac{2ab}{a+b}-a} + \frac{1}{\frac{2ab}{a+b}-b}$
$= \frac{a+b}{2ab-a^{2}-ab}+\frac{a+b}{2ab-ab-b^{2}} $
$= \frac{a+b}{ab-a^{2}}+ \frac{a+b}{ab-b^{2}} $
$= a+b\left[\frac{1}{a\left(b-c\right)} + \frac{2}{b\left(a-b\right)}\right] $
$ = \left(a+b\right)\left[\frac{a-b}{ab\left(a-b\right)}\right] $
$ = \frac{a+b}{ab} = \frac{1}{a}+\frac{1}{b}$