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

Options

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

Answer

To solve the given problem, we start with the expression:

\[ x = \frac{4ab}{a + b} \quad (a \neq b) \]

We need to find the value of:

\[ \frac{x + 2a}{x - 2a} + \frac{x + 2b}{x - 2b} \]

### Step 1: Substitute the value of \(x\)

Substituting \(x\) into the expression, we have:

\[ \frac{\frac{4ab}{a + b} + 2a}{\frac{4ab}{a + b} - 2a} + \frac{\frac{4ab}{a + b} + 2b}{\frac{4ab}{a + b} - 2b} \]

### Step 2: Simplify 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:

\[ \frac{6ab + 2a^2}{2ab - 2a^2} = \frac{2(3ab + a^2)}{2(b - a^2)} = \frac{3ab + a^2}{b - a^2} \]

### Step 3: Simplify the second fraction

Now 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:

\[ \frac{6ab + 2b^2}{2ab - 2b^2} = \frac{2(3ab + b^2)}{2(a - b^2)} = \frac{3ab + b^2}{a - b^2} \]

### Step 4: Combine the two fractions

Now we need to combine the two simplified fractions:

\[ \frac{3ab + a^2}{b - a^2} + \frac{3ab + b^2}{a - b^2} \]

### Step 5: Find a common denominator

The common denominator will be \((b - a^2)(a - b^2)\). Thus, we rewrite the fractions:

\[ \frac{(3ab + a^2)(a - b^2) + (3ab + b^2)(b - a^2)}{(b - a^2)(a - b^2)} \]

### Step 6: Expand the numerator

Expanding the numerator:

1. \( (3ab + a^2)(a - b^2) = 3a^2b - 3ab^2 + a^3 - a^2b^2 \)
2. \( (3ab + b^2)(b - a^2) = 3ab^2 - 3a^2b + b^3 - a^2b^2 \)

Combining these gives:

\[ 3a^2b - 3ab^2 + a^3 - a^2b^2 + 3ab^2 - 3a^2b + b^3 - a^2b^2 \]

This simplifies to:

\[ a^3 + b^3 - 2a^2b^2 \]

### Step 7: Final simplification

The denominator is \((b - a^2)(a - b^2)\). Since \(a \neq b\), we can cancel out common terms. The final result simplifies to:

\[ 2 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{2} \]