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

Putting $H=\frac{2ab}{a+b},$ we have $\frac{1}{H-a}+\frac{1}{H-b}$
$=\frac{1}{(\frac{2ab}{a+b}-a)}+\frac{1}{(\frac{2ab}{a+b}-b)}=\frac{a+b}{ab-a^2}+\frac{a+b}{ab-b^2}$
$=\left(\frac{a+b}{b-a}\right)(\frac{1}{a}-\frac{1}{b})=\left(\frac{a+b}{b-a}\right)\left(\frac{b-a}{ab}\right)=\frac{a+b}{ab}=\frac{1}{a}+\frac{1}{b}.$