If `f(x+2y , x-2y)=x y ,` 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 need to find the function \( f(x, y) \) given that \( f(x + 2y, x - 2y) = xy \).

### Step-by-step Solution:

1. **Substitution of Variables**:
Let \( u = x + 2y \) and \( v = x - 2y \). This will help us express \( x \) and \( y \) in terms of \( u \) and \( v \).

2. **Adding the Equations**:
Add the equations for \( u \) and \( v \):
\[ u + v = (x + 2y) + (x - 2y) = 2x \]
Therefore, we can express \( x \) as:
\[ x = \frac{u + v}{2} \]

3. **Subtracting the Equations**:
Now, subtract the equation for \( v \) from \( u \):
\[ u - v = (x + 2y) - (x - 2y) = 4y \]
Thus, we can express \( y \) as:
\[ y = \frac{u - v}{4} \]

4. **Expressing the Function**:
Now, substitute \( x \) and \( y \) back into the equation \( f(u, v) = xy \):
\[ f(u, v) = \left(\frac{u + v}{2}\right) \left(\frac{u - v}{4}\right) \]

5. **Simplifying the Expression**:
Multiply the two fractions:
\[ f(u, v) = \frac{(u + v)(u - v)}{8} = \frac{u^2 - v^2}{8} \]

6. **Replacing Back to Original Variables**:
Finally, replace \( u \) and \( v \) back with \( x \) and \( y \):
\[ f(x, y) = \frac{x^2 - y^2}{8} \]

### Final Answer:
Thus, the function \( f(x, y) \) is:
\[ f(x, y) = \frac{x^2 - y^2}{8} \]