If `m=(4pq)/(p+q)` then the value of `(m+2p)/(m-2p)+(m+2q)/(m-2q)`

Options

A. 2
B. 4
C. `(2mpq)/((p+q))`
D. None of these

Answer

To solve the problem, we start with the given expression for \( m \):

\[ m = \frac{4pq}{p + q} \]

We need to find the value of

\[ \frac{m + 2p}{m - 2p} + \frac{m + 2q}{m - 2q} \]

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

Substituting \( m \) into the expression:

\[ \frac{\frac{4pq}{p + q} + 2p}{\frac{4pq}{p + q} - 2p} + \frac{\frac{4pq}{p + q} + 2q}{\frac{4pq}{p + q} - 2q} \]

### Step 2: Simplify the first fraction

For the first fraction:

\[ \frac{m + 2p}{m - 2p} = \frac{\frac{4pq}{p + q} + 2p}{\frac{4pq}{p + q} - 2p} \]

To simplify, we can find a common denominator for the numerator and the denominator:

\[ = \frac{\frac{4pq + 2p(p + q)}{p + q}}{\frac{4pq - 2p(p + q)}{p + q}} = \frac{4pq + 2p^2 + 2pq}{4pq - 2p^2 - 2pq} \]

This simplifies to:

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

### Step 3: Simplify the second fraction

Now for the second fraction:

\[ \frac{m + 2q}{m - 2q} = \frac{\frac{4pq}{p + q} + 2q}{\frac{4pq}{p + q} - 2q} \]

Using a similar approach:

\[ = \frac{\frac{4pq + 2q(p + q)}{p + q}}{\frac{4pq - 2q(p + q)}{p + q}} = \frac{4pq + 2pq + 2q^2}{4pq - 2pq - 2q^2} \]

This simplifies to:

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

### Step 4: Add the two fractions

Now we add the two simplified 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: Simplify the numerator

Expanding the numerator:

1. First term:
\[ (3pq + p^2)(pq - q^2) = 3p^2q - 3pq^3 + p^2pq - p^2q^2 \]

2. Second term:
\[ (3pq + q^2)(pq - p^2) = 3pq^2 - 3p^2q + q^2pq - q^2p^2 \]

Combining these gives:

\[ = 3p^2q + p^2pq - 3pq^3 - p^2q^2 + 3pq^2 - 3p^2q + q^2pq - q^2p^2 \]

### Step 6: Final simplification

Combining like terms leads to:

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

### Conclusion

After simplifying, we find that the final result is:

\[ = 2 \]

Thus, the value of

\[ \frac{m + 2p}{m - 2p} + \frac{m + 2q}{m - 2q} = 2 \]