`{:("Column A",a > b > 0,"Column B),((a - b)/(a + b),,(a^2 - b^2)/(a^2 + b^2)):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to compare the expressions in Column A and Column B given that \( a > b > 0 \).

### Step-by-Step Solution:

1. **Identify the Expressions**:
- Column A: \( \frac{a - b}{a + b} \)
- Column B: \( \frac{a^2 - b^2}{a^2 + b^2} \)

2. **Simplify Column A**:
- We can rewrite Column A's expression:
\[ \frac{a - b}{a + b} = \frac{(a - b)(a + b)}{(a + b)(a + b)} = \frac{a^2 - b^2}{(a + b)^2} \]
- Here, we used the identity \( a^2 - b^2 = (a - b)(a + b) \).

3. **Rewrite Column B**:
- Column B is already given as:
\[ \frac{a^2 - b^2}{a^2 + b^2} \]

4. **Compare the Two Columns**:
- Now we have:
- Column A: \( \frac{a^2 - b^2}{(a + b)^2} \)
- Column B: \( \frac{a^2 - b^2}{a^2 + b^2} \)

5. **Analyze the Numerators**:
- Both columns have the same numerator \( a^2 - b^2 \). Since \( a > b \), \( a^2 - b^2 > 0 \).

6. **Compare the Denominators**:
- For Column A, the denominator is \( (a + b)^2 \).
- For Column B, the denominator is \( a^2 + b^2 \).
- Since \( a \) and \( b \) are both positive, we know that \( (a + b)^2 > a^2 + b^2 \) because:
\[ (a + b)^2 = a^2 + 2ab + b^2 > a^2 + b^2 \]
- Thus, \( (a + b)^2 > a^2 + b^2 \).

7. **Conclusion**:
- Since both columns have the same positive numerator and Column A has a larger denominator than Column B, it follows that:
\[ \frac{a^2 - b^2}{(a + b)^2} < \frac{a^2 - b^2}{a^2 + b^2} \]
- Therefore, Column B is larger than Column A.

### Final Answer:
- **Column B is larger**.