If `x=(4ab)/(a+b)`, then the value of `(x+2a)/(x-2a)+(x+2b)/(x-2b)` is

Options

A. a
B. b
C. 0
D. 2

Answer

To solve the problem, we start with the given expression and substitute the value of \( x \).

### Step-by-Step Solution:

1. **Given Expression**:
\[ x = \frac{4ab}{a + b} \]
We need to evaluate:
\[ \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \]

2. **Substituting \( x \)**:
Substitute \( x \) into the expression:
\[ \frac{\frac{4ab}{a + b} + 2a}{\frac{4ab}{a + b} - 2a} + \frac{\frac{4ab}{a + b} + 2b}{\frac{4ab}{a + b} - 2b} \]

3. **Simplifying the First Fraction**:
For the first fraction:
\[ \frac{\frac{4ab + 2a(a + b)}{a + b}}{\frac{4ab - 2a(a + b)}{a + b}} = \frac{4ab + 2a^2 + 2ab}{4ab - 2a^2 - 2ab} \]
This simplifies to:
\[ \frac{6ab + 2a^2}{2ab - 2a^2} = \frac{2(3ab + a^2)}{2(ab - a^2)} = \frac{3ab + a^2}{ab - a^2} \]

4. **Simplifying the Second Fraction**:
For the second fraction:
\[ \frac{\frac{4ab + 2b(a + b)}{a + b}}{\frac{4ab - 2b(a + b)}{a + b}} = \frac{4ab + 2b^2 + 2ab}{4ab - 2b^2 - 2ab} \]
This simplifies to:
\[ \frac{6ab + 2b^2}{2ab - 2b^2} = \frac{2(3ab + b^2)}{2(ab - b^2)} = \frac{3ab + b^2}{ab - b^2} \]

5. **Combining Both Fractions**:
Now we combine both simplified fractions:
\[ \frac{3ab + a^2}{ab - a^2} + \frac{3ab + b^2}{ab - b^2} \]
To combine, we find a common denominator:
\[ = \frac{(3ab + a^2)(ab - b^2) + (3ab + b^2)(ab - a^2)}{(ab - a^2)(ab - b^2)} \]

6. **Expanding the Numerator**:
Expanding both terms in the numerator:
\[ = \frac{(3ab^2 - 3ab^3 + a^2b - a^2b^2) + (3a^2b - 3a^2b^2 + b^2a - b^2a^2)}{(ab - a^2)(ab - b^2)} \]

7. **Final Simplification**:
After simplifying the numerator, we can find that:
\[ = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]