If H be the H.M. between a and b, then the value of `(H)/(a)+(H)/(b)` is

Options

A. 2
B. `(ab)/(a+b)`
C. `(a+b)/(ab)`
D. none of these

Answer

To solve the problem, we need to find the value of \(\frac{H}{a} + \frac{H}{b}\) where \(H\) is the harmonic mean between \(a\) and \(b\).

### Step-by-Step Solution:

1. **Definition of Harmonic Mean**:
The harmonic mean \(H\) of two numbers \(a\) and \(b\) is given by the formula:
\[ H = \frac{2ab}{a + b} \]

2. **Calculate \(\frac{H}{a}\)**:
We start by substituting the value of \(H\) into \(\frac{H}{a}\):
\[ \frac{H}{a} = \frac{\frac{2ab}{a + b}}{a} = \frac{2ab}{a(a + b)} = \frac{2b}{a + b} \]

3. **Calculate \(\frac{H}{b}\)**:
Next, we calculate \(\frac{H}{b}\):
\[ \frac{H}{b} = \frac{\frac{2ab}{a + b}}{b} = \frac{2ab}{b(a + b)} = \frac{2a}{a + b} \]

4. **Combine \(\frac{H}{a}\) and \(\frac{H}{b}\)**:
Now, we add \(\frac{H}{a}\) and \(\frac{H}{b}\):
\[ \frac{H}{a} + \frac{H}{b} = \frac{2b}{a + b} + \frac{2a}{a + b} \]

5. **Simplifying the Expression**:
Since both fractions have the same denominator, we can combine them:
\[ \frac{H}{a} + \frac{H}{b} = \frac{2b + 2a}{a + b} = \frac{2(a + b)}{a + b} \]

6. **Final Result**:
The \(a + b\) in the numerator and denominator cancels out:
\[ \frac{H}{a} + \frac{H}{b} = 2 \]

Thus, the value of \(\frac{H}{a} + \frac{H}{b}\) is \(2\).