Question

$H_1,H_2$ are 2 H.M.’s between a,b then $\frac{H_1+H_2}{H_1.H_2}=$

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

Answer 1

Answer: (B)

$\frac{a+b}{ab}$

Given,$H_1{H}_2$ be the two H.M's
and lets say $a,H_1,H_2,b$ be is a harmonic progression.

And $\frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$in corresponding A.P after taking reciprocals.
And we know that in A.P. $r=T_2-T_1=T_3-T_2....$

$\Rightarrow \frac{1}{H_1}-\frac{1}{a}=\frac{1}{H_2}-\frac{1}{H_1}=\frac{1}{b}-\frac{1}{H_2}\Rightarrow \frac{1}{H_1}+\frac{1}{H_2}=\frac{1}{b}+\frac{1}{a}\Rightarrow \frac{H_1+H_2}{H_1.H_2}=\frac{a+b}{ab}$

$\therefore \frac{H_1.H_2}{H_1+H_2}=\frac{ab}{a+b}$

Hence, option $\left(C\right)$ is the correct answer$