The simplified value of
`(1- ( 2xy )/( x ^(2) + y ^(2))) div ((x ^(3) - y ^(3))/( x - y ) - 3 xy )` is

Options

A. `(1)/( x ^(2) - y ^(2))`
B. `(1)/(x ^(2) + y ^(2))`
C. `(1)/(x -y)`
D. `(1)/( x + y)`

Answer

To simplify the expression

\[ \frac{1 - \frac{2xy}{x^2 + y^2}}{\frac{x^3 - y^3}{x - y} - 3xy} \]

we will follow these steps:

### Step 1: Simplify the numerator
The numerator is

\[ 1 - \frac{2xy}{x^2 + y^2} \]

To combine these terms, we need a common denominator, which is \(x^2 + y^2\):

\[ 1 = \frac{x^2 + y^2}{x^2 + y^2} \]

Thus, we can rewrite the numerator:

\[ \frac{x^2 + y^2 - 2xy}{x^2 + y^2} \]

### Step 2: Simplify the denominator
The denominator is

\[ \frac{x^3 - y^3}{x - y} - 3xy \]

Using the difference of cubes formula, we know that

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

So we can rewrite the first term:

\[ \frac{(x - y)(x^2 + xy + y^2)}{x - y} = x^2 + xy + y^2 \]

Now, substituting this back into the denominator gives us:

\[ x^2 + xy + y^2 - 3xy = x^2 - 2xy + y^2 \]

### Step 3: Combine the simplified numerator and denominator
Now we have:

\[ \frac{\frac{x^2 + y^2 - 2xy}{x^2 + y^2}}{x^2 - 2xy + y^2} \]

The expression simplifies to:

\[ \frac{x^2 + y^2 - 2xy}{(x^2 + y^2)(x^2 - 2xy + y^2)} \]

### Step 4: Recognize the numerator
Notice that \(x^2 + y^2 - 2xy\) can be factored as:

\[ (x - y)^2 \]

So we can rewrite our expression as:

\[ \frac{(x - y)^2}{(x^2 + y^2)(x^2 - 2xy + y^2)} \]

### Step 5: Simplify the denominator further
The term \(x^2 - 2xy + y^2\) can also be factored as \((x - y)^2\). Thus, we have:

\[ \frac{(x - y)^2}{(x^2 + y^2)(x - y)^2} \]

### Step 6: Cancel out common terms
The \((x - y)^2\) terms in the numerator and denominator cancel out, leaving us with:

\[ \frac{1}{x^2 + y^2} \]

### Final Answer
Thus, the simplified value of the expression is:

\[ \frac{1}{x^2 + y^2} \]