If H be the harmonic mean of a and b then find the value of `H/a+H/b-2`

Answer

To solve the problem, we need to find the value of \( \frac{H}{a} + \frac{H}{b} - 2 \), where \( H \) is the harmonic mean of \( a \) and \( b \).

### Step-by-Step Solution:

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

2. **Substitute \( H \) into the expression**:
We need to evaluate:
\[ \frac{H}{a} + \frac{H}{b} - 2 \]
Substituting the expression for \( H \):
\[ \frac{H}{a} = \frac{2ab}{a + b} \cdot \frac{1}{a} = \frac{2b}{a + b} \]
\[ \frac{H}{b} = \frac{2ab}{a + b} \cdot \frac{1}{b} = \frac{2a}{a + b} \]

3. **Combine the two fractions**:
Now, we can add \( \frac{H}{a} \) and \( \frac{H}{b} \):
\[ \frac{H}{a} + \frac{H}{b} = \frac{2b}{a + b} + \frac{2a}{a + b} = \frac{2b + 2a}{a + b} = \frac{2(a + b)}{a + b} \]

4. **Simplify the expression**:
The expression simplifies to:
\[ \frac{2(a + b)}{a + b} = 2 \]

5. **Subtract 2 from the result**:
Now, we substitute this back into our original expression:
\[ \frac{H}{a} + \frac{H}{b} - 2 = 2 - 2 = 0 \]

### Final Answer:
Thus, the value of \( \frac{H}{a} + \frac{H}{b} - 2 \) is:
\[ \boxed{0} \]