If `(a+b)x =a and (a+b) y = b` then the value of `(x^(2)+y^(2))/(x^(2)-y^(2))` is

Options

A. `(a^(2)-b^(2))/(a^(2)+b^(2))`
B. `(a^(2))/(a^(2)+b^(2))`
C. `(b^(2))/(a^(2) +b^(2))`
D. `(a^(2) +b^(2))/(a^(2)-b^(2))`

Answer

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

1. \((a + b)x = a\)
2. \((a + b)y = b\)

### Step 1: Solve for \(x\) and \(y\)

From the first equation, we can isolate \(x\):

\[ x = \frac{a}{a + b} \]

From the second equation, we can isolate \(y\):

\[ y = \frac{b}{a + b} \]

### Step 2: Find \(x^2\) and \(y^2\)

Now, we calculate \(x^2\) and \(y^2\):

\[ x^2 = \left(\frac{a}{a + b}\right)^2 = \frac{a^2}{(a + b)^2} \]

\[ y^2 = \left(\frac{b}{a + b}\right)^2 = \frac{b^2}{(a + b)^2} \]

### Step 3: Substitute \(x^2\) and \(y^2\) into the expression

We need to evaluate the expression:

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

Substituting the values of \(x^2\) and \(y^2\):

\[ \frac{\frac{a^2}{(a + b)^2} + \frac{b^2}{(a + b)^2}}{\frac{a^2}{(a + b)^2} - \frac{b^2}{(a + b)^2}} \]

### Step 4: Simplify the expression

The numerator simplifies to:

\[ \frac{a^2 + b^2}{(a + b)^2} \]

The denominator simplifies to:

\[ \frac{a^2 - b^2}{(a + b)^2} \]

Now we can rewrite the entire expression:

\[ \frac{\frac{a^2 + b^2}{(a + b)^2}}{\frac{a^2 - b^2}{(a + b)^2}} = \frac{a^2 + b^2}{a^2 - b^2} \]

### Final Result

Thus, the value of \(\frac{x^2 + y^2}{x^2 - y^2}\) is:

\[ \frac{a^2 + b^2}{a^2 - b^2} \]