Question

If $H$ is the harmonic mean between $P$ and $Q$, then the value of $\frac{H}{P}+\frac{H}{Q}$ is

• A. $2$
• B. $\frac{PQ}{P+Q}$
• C. $\frac{1}{2}$
• D. $\frac{P+Q}{PQ}$

Answer 1

The correct option is A

$2$

Explanation for the correct option:

Finding the value of expression:

Since, $P$ and $Q$ are in HP, then

$$\begin{gathered} H=\frac{2PQ}{P+Q}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{H}\mathbf{M}\mathbf{=}\frac{\mathbf{2}\mathbf{a}\mathbf{b}}{\mathbf{a}\mathbf{+}\mathbf{b}}\mathbf{]} \\ \Rightarrow \frac{H}{P}=\frac{2Q}{P+Q}...\left(1\right) \\ and\frac{H}{Q}=\frac{2P}{P+Q}...\left(2\right) \end{gathered}$$

Add equation $\left(1\right)$ and $\left(2\right)$, we get

$\begin{array}{rcl}\frac{H}{P}+\frac{H}{Q}& =& \frac{2Q}{P+Q}+\frac{2P}{P+Q}\\ & =& \frac{2\left(P+Q\right)}{P+Q}\\ & =& 2\end{array}$

Hence, the correct option is A.