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

Options

A. x
B. y
C. 0
D. none of these

Explanation

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

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

We need to find the value of:

\[ \frac{f(x,y) + f(y,x)}{2} \]

### Step 1: Change Variables
Let’s define new variables:
- Let \( p = x + y \)
- Let \( q = x - y \)

From these equations, we can express \( x \) and \( y \) in terms of \( p \) and \( q \):

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

### Step 2: Express \( f(p, q) \)
Now, we can express \( f(p, q) \) in terms of \( p \) and \( q \):

\[ f(p, q) = xy = \left(\frac{p + q}{2}\right)\left(\frac{p - q}{2}\right) \]

### Step 3: Simplify \( f(p, q) \)
Now, we simplify \( f(p, q) \):

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

### Step 4: Find \( f(y, x) \)
Next, we need to find \( f(y, x) \). We can switch \( x \) and \( y \):

\[ f(y, x) = f\left(\frac{p - q}{2}, \frac{p + q}{2}\right) = \frac{\left(\frac{p - q}{2}\right)\left(\frac{p + q}{2}\right)}{4} \]

### Step 5: Simplify \( f(y, x) \)
Now, we simplify \( f(y, x) \):

\[ f(y, x) = \frac{(p - q)(p + q)}{4} = \frac{p^2 - q^2}{4} \]

### Step 6: Add \( f(x, y) \) and \( f(y, x) \)
Now we can add \( f(x, y) \) and \( f(y, x) \):

\[ f(x, y) + f(y, x) = \frac{p^2 - q^2}{4} + \frac{p^2 - q^2}{4} = \frac{2(p^2 - q^2)}{4} = \frac{p^2 - q^2}{2} \]

### Step 7: Final Calculation
Now we can find the final result:

\[ \frac{f(x, y) + f(y, x)}{2} = \frac{\frac{p^2 - q^2}{2}}{2} = \frac{p^2 - q^2}{4} \]

### Conclusion
Thus, the final answer is:

\[ \frac{f(x, y) + f(y, x)}{2} = \frac{(x+y)^2 - (x-y)^2}{4} \]