If ` x = ( 6 pq)/( p + q)` then the vaue of `( x + 3 p)/( x - 3 p) + ( x + 3 q)/( x - 3q)` is

Options

A. 6
B. 8
C. 2
D. 3

Answer

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

\[ x = \frac{6pq}{p + q} \]

We need to find the value of:

\[ \frac{x + 3p}{x - 3p} + \frac{x + 3q}{x - 3q} \]

### Step 1: Substitute \( x \)

First, we substitute \( x \) in the expression:

\[ \frac{\frac{6pq}{p + q} + 3p}{\frac{6pq}{p + q} - 3p} + \frac{\frac{6pq}{p + q} + 3q}{\frac{6pq}{p + q} - 3q} \]

### Step 2: Simplify the first fraction

For the first fraction:

\[ \frac{\frac{6pq + 3p(p + q)}{p + q}}{\frac{6pq - 3p(p + q)}{p + q}} = \frac{6pq + 3p^2 + 3pq}{6pq - 3p^2 - 3pq} \]

This simplifies to:

\[ \frac{9pq + 3p^2}{3pq - 3p^2} = \frac{3(3pq + p^2)}{3(pq - p^2)} = \frac{3pq + p^2}{pq - p^2} \]

### Step 3: Simplify the second fraction

Now for the second fraction:

\[ \frac{\frac{6pq + 3q(p + q)}{p + q}}{\frac{6pq - 3q(p + q)}{p + q}} = \frac{6pq + 3q^2 + 3pq}{6pq - 3q^2 - 3pq} \]

This simplifies to:

\[ \frac{9pq + 3q^2}{3pq - 3q^2} = \frac{3(3pq + q^2)}{3(pq - q^2)} = \frac{3pq + q^2}{pq - q^2} \]

### Step 4: Combine the two fractions

Now we combine the two fractions:

\[ \frac{3pq + p^2}{pq - p^2} + \frac{3pq + q^2}{pq - q^2} \]

To add these fractions, we need a common denominator:

\[ = \frac{(3pq + p^2)(pq - q^2) + (3pq + q^2)(pq - p^2)}{(pq - p^2)(pq - q^2)} \]

### Step 5: Expand the numerator

Expanding the numerator gives:

\[ (3pq + p^2)(pq - q^2) + (3pq + q^2)(pq - p^2) \]

This will yield a polynomial in \( p \) and \( q \).

### Step 6: Simplify the numerator

After simplifying the numerator, we notice that terms will cancel out, leading us to a simplified expression.

### Final Result

After completing the simplification, we find that the expression evaluates to:

\[ 2 \]