If `2apq = (p + q)^(2) - (p - q) ^(2)` , then the value of a is

Options

A. 2
B. 1
C. 4
D. 8

Answer

To solve the equation \( 2apq = (p + q)^{2} - (p - q)^{2} \), we will follow these steps:

### Step 1: Expand the squares on the right-hand side
We start by expanding \( (p + q)^{2} \) and \( (p - q)^{2} \).

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

### Step 2: Substitute the expansions back into the equation
Now we substitute these expansions back into the equation:

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

### Step 3: Simplify the right-hand side
Next, we simplify the right-hand side:

\[ 2apq = p^{2} + 2pq + q^{2} - p^{2} + 2pq - q^{2} \]

Notice that \( p^{2} \) and \( -p^{2} \) cancel out, and \( q^{2} \) and \( -q^{2} \) also cancel out:

\[ 2apq = 2pq + 2pq \]

This simplifies to:

\[ 2apq = 4pq \]

### Step 4: Divide both sides by \( 2pq \)
Assuming \( pq \neq 0 \), we can divide both sides by \( 2pq \):

\[ a = \frac{4pq}{2pq} \]

### Step 5: Simplify the fraction
This simplifies to:

\[ a = 2 \]

### Conclusion
Thus, the value of \( a \) is \( 2 \).

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