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)

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