If ` x = ( 8ab)/(a + b) (a ne b)` then the value of `( x + 4a)/( x - 4a) + ( x + 4b)/( x - 4b)` is

Options

A. 0
B. 1
C. 2
D. 4

Answer

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

\[ x = \frac{8ab}{a + b} \quad (a \neq b) \]

We need to find the value of:

\[ \frac{x + 4a}{x - 4a} + \frac{x + 4b}{x - 4b} \]

### Step 1: Rewrite the expression

We can rewrite the expression as:

\[ \frac{x + 4a}{x - 4a} + \frac{x + 4b}{x - 4b} \]

### Step 2: Apply the component and dividend method

Using the component and dividend method, we can express:

\[ \frac{x + 4a}{x - 4a} = \frac{(x + 4a) + (x - 4a)}{(x + 4a) - (x - 4a)} = \frac{2x}{8a} = \frac{x}{4a} \]

Similarly, for the second part:

\[ \frac{x + 4b}{x - 4b} = \frac{(x + 4b) + (x - 4b)}{(x + 4b) - (x - 4b)} = \frac{2x}{8b} = \frac{x}{4b} \]

### Step 3: Combine the two fractions

Now we can combine the two fractions:

\[ \frac{x + 4a}{x - 4a} + \frac{x + 4b}{x - 4b} = \frac{x}{4a} + \frac{x}{4b} \]

### Step 4: Find a common denominator

The common denominator for \(4a\) and \(4b\) is \(4ab\):

\[ \frac{x}{4a} + \frac{x}{4b} = \frac{bx + ax}{4ab} = \frac{x(a + b)}{4ab} \]

### Step 5: Substitute the value of \(x\)

Now substitute \(x = \frac{8ab}{a + b}\):

\[ \frac{\frac{8ab}{a + b}(a + b)}{4ab} = \frac{8ab}{4ab} = 2 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{2} \]