If `f (x+2y, x-2y) = xy, `then f (x, y) equals

Options

A. `(x^2 -y^2 )/( 8 )`
B. ` (x^2 - y^2)/(4)`
C. ` (x^2 +y^2)/(4)`
D. `(x^2 -y^2)/(2)`

Answer

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

\[ f(x + 2y, x - 2y) = xy \]

We need to find \( f(x, y) \).

### Step 1: Define new variables
Let:
- \( u = x + 2y \)
- \( v = x - 2y \)

### Step 2: Express \( x \) and \( y \) in terms of \( u \) and \( v \)
To find \( x \) and \( y \) in terms of \( u \) and \( v \):

1. **Add \( u \) and \( v \)**:
\[ u + v = (x + 2y) + (x - 2y) = 2x \quad \Rightarrow \quad x = \frac{u + v}{2} \]

2. **Subtract \( v \) from \( u \)**:
\[ u - v = (x + 2y) - (x - 2y) = 4y \quad \Rightarrow \quad y = \frac{u - v}{4} \]

### Step 3: Substitute \( x \) and \( y \) into the right-hand side
Now, we substitute \( x \) and \( y \) back into the equation \( f(u, v) = xy \):

\[ xy = \left(\frac{u + v}{2}\right) \left(\frac{u - v}{4}\right) \]

### Step 4: Simplify the expression
Now, simplify the right-hand side:

\[ xy = \frac{(u + v)(u - v)}{8} = \frac{u^2 - v^2}{8} \]

Thus, we have:

\[ f(u, v) = \frac{u^2 - v^2}{8} \]

### Step 5: Replace \( u \) and \( v \) with \( x \) and \( y \)
Now, we need to find \( f(x, y) \):

Replace \( u \) with \( x \) and \( v \) with \( y \):

\[ f(x, y) = \frac{x^2 - y^2}{8} \]

### Final Answer
Thus, the function \( f(x, y) \) is:

\[ f(x, y) = \frac{x^2 - y^2}{8} \]

---