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

Answer

To find the value of \( \frac{H}{a} + \frac{H}{b} - 2 \) where \( H \) is the harmonic mean of \( a \) and \( b \), we can follow these steps:

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

### Step 2: Substitute \( H \) into the expression
We need to evaluate:
\[ \frac{H}{a} + \frac{H}{b} - 2 \]
Substituting the expression for \( H \):
\[ \frac{H}{a} = \frac{2ab}{a(a + b)} = \frac{2b}{a + b} \]
\[ \frac{H}{b} = \frac{2ab}{b(a + b)} = \frac{2a}{a + b} \]
Now, substituting these into the expression:
\[ \frac{H}{a} + \frac{H}{b} = \frac{2b}{a + b} + \frac{2a}{a + b} \]

### Step 3: Combine the fractions
Since both fractions have the same denominator, we can combine them:
\[ \frac{H}{a} + \frac{H}{b} = \frac{2b + 2a}{a + b} = \frac{2(a + b)}{a + b} \]

### Step 4: Simplify the expression
Now, we can simplify:
\[ \frac{H}{a} + \frac{H}{b} = 2 \]

### Step 5: Substitute back into the original expression
Now, substituting back into the 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:
\[ \boxed{0} \]