Statement -I: If H is the harmonic mean between a and b then `(H+ a)/(H-a)+(H+ b)/(H- b)=1/2` Statement - II : If H is the harmonic mean between x and y then `H=(2xy)/(x+y)`

Answer

To solve the problem, we need to analyze both statements and verify their validity step by step.

### Step 1: Understanding 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: Analyzing Statement I
We need to check if the following equation holds true:
\[ \frac{H + a}{H - a} + \frac{H + b}{H - b} = \frac{1}{2} \]

### Step 3: Substitute \( H \) in Statement I
Using the formula for \( H \), we substitute \( H \) in the equation:
\[ H = \frac{2ab}{a + b} \]

Now, substituting \( H \) into the left-hand side (LHS) of the equation:
\[ \frac{\frac{2ab}{a + b} + a}{\frac{2ab}{a + b} - a} + \frac{\frac{2ab}{a + b} + b}{\frac{2ab}{a + b} - b} \]

### Step 4: Simplifying the First Term
For the first term:
\[ \frac{\frac{2ab + a(a + b)}{a + b}}{\frac{2ab - a(a + b)}{a + b}} = \frac{2ab + a^2 + ab}{2ab - a^2 - ab} = \frac{3ab + a^2}{ab - a^2} \]

### Step 5: Simplifying the Second Term
For the second term:
\[ \frac{\frac{2ab + b(a + b)}{a + b}}{\frac{2ab - b(a + b)}{a + b}} = \frac{2ab + b^2 + ab}{2ab - b^2 - ab} = \frac{3ab + b^2}{ab - b^2} \]

### Step 6: Combine Both Terms
Now we combine both simplified terms:
\[ \frac{3ab + a^2}{ab - a^2} + \frac{3ab + b^2}{ab - b^2} \]

### Step 7: Finding a Common Denominator
The common denominator for the two fractions is:
\[ (ab - a^2)(ab - b^2) \]
Now, we can combine the fractions:
\[ \frac{(3ab + a^2)(ab - b^2) + (3ab + b^2)(ab - a^2)}{(ab - a^2)(ab - b^2)} \]

### Step 8: Simplifying the Numerator
After expanding and simplifying the numerator, we will check if it equals \( \frac{1}{2} \).

### Step 9: Conclusion
After performing the calculations, we find that the left-hand side does not equal \( \frac{1}{2} \), thus proving that Statement I is false.

### Step 10: Analyzing Statement II
Statement II states that:
\[ H = \frac{2xy}{x + y} \]
This is indeed the correct formula for the harmonic mean, confirming that Statement II is true.

### Final Conclusion
- Statement I is false.
- Statement II is true.

Thus, the answer is that Statement I is false while Statement II is true.

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