Question

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

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

Answer 1

The correct option is C $\frac{1}{a}+\frac{1}{b}$

$\because H$ is harmonic mean between $a$ and $b$
$\therefore H=\frac{2ab}{a+b}$
Therefore, $\frac{1}{H-a}+\frac{1}{H-b}=\frac{1}{\frac{2ab}{a+b}-a}+\frac{1}{\frac{2ab}{a+b}-b}$
$=\frac{a+b}{ab-a^2}+\frac{a+b}{ab-b^2}=\frac{a+b}{b-a}(\frac{1}{a}-\frac{1}{b})=\frac{a+b}{ab}=\frac{1}{a}+\frac{1}{b}$
Hence, option 'C' is correct.