If f (x + ay , x - ay) = axy , then f(x,y) is equal to

Options

A. xy
B. ` x^2 + a^2 y^2`
C. ` (x^2 -y^2)/(4)`
D. none of these

Answer

To solve the problem \( f(x + ay, x - ay) = axy \) and find \( f(x, y) \), we will follow these steps:

### Step 1: Define new variables
Let:
- \( u = x + ay \)
- \( v = x - ay \)

### Step 2: Express \( x \) and \( y \) in terms of \( u \) and \( v \)
From the definitions of \( u \) and \( v \):
1. Adding the two equations:
\[ u + v = (x + ay) + (x - ay) = 2x \implies x = \frac{u + v}{2} \]

2. Subtracting the second equation from the first:
\[ u - v = (x + ay) - (x - ay) = 2ay \implies y = \frac{u - v}{2a} \]

### Step 3: Substitute \( x \) and \( y \) into the original function
Now we substitute \( x \) and \( y \) back into the equation \( f(u, v) = axy \):
\[ f(u, v) = a \left(\frac{u + v}{2}\right) \left(\frac{u - v}{2a}\right) \]

### Step 4: Simplify the expression
Simplifying the right-hand side:
\[ f(u, v) = a \cdot \frac{(u + v)(u - v)}{4a} = \frac{(u + v)(u - v)}{4} \]
Using the difference of squares:
\[ f(u, v) = \frac{u^2 - v^2}{4} \]

### Step 5: Replace \( u \) and \( v \) back with \( x \) and \( y \)
Now, substitute back \( u = x + ay \) and \( v = x - ay \):
\[ f(x, y) = \frac{(x + ay)^2 - (x - ay)^2}{4} \]

### Step 6: Expand the squares
\[ f(x, y) = \frac{(x^2 + 2axy + a^2y^2) - (x^2 - 2axy + a^2y^2)}{4} \]
This simplifies to:
\[ f(x, y) = \frac{4axy}{4} = axy \]

### Final Result
Thus, we find that:
\[ f(x, y) = axy \]