If x = `(b)/(a - b) and y = (b)/(a + b)` then the value of `(1)/(x) + (1)/(y)` is

Options

A. `(a^(2) + b^(2))/(ab)`
B. `(b^(2) - a^(2))/(ab)`
C. `(a^(2) - b^(2))/(ab)`
D. None of these

Answer

To solve the problem, we need to find the value of \( \frac{1}{x} + \frac{1}{y} \) given that \( x = \frac{b}{a - b} \) and \( y = \frac{b}{a + b} \).

### Step-by-Step Solution:

1. **Find \( \frac{1}{x} \)**:
\[ x = \frac{b}{a - b} \]
Therefore,
\[ \frac{1}{x} = \frac{1}{\frac{b}{a - b}} = \frac{a - b}{b} \]

2. **Find \( \frac{1}{y} \)**:
\[ y = \frac{b}{a + b} \]
Therefore,
\[ \frac{1}{y} = \frac{1}{\frac{b}{a + b}} = \frac{a + b}{b} \]

3. **Add \( \frac{1}{x} \) and \( \frac{1}{y} \)**:
\[ \frac{1}{x} + \frac{1}{y} = \frac{a - b}{b} + \frac{a + b}{b} \]

4. **Combine the fractions**:
Since both fractions have the same denominator, we can combine them:
\[ \frac{1}{x} + \frac{1}{y} = \frac{(a - b) + (a + b)}{b} \]

5. **Simplify the numerator**:
\[ (a - b) + (a + b) = a - b + a + b = 2a \]
Therefore,
\[ \frac{1}{x} + \frac{1}{y} = \frac{2a}{b} \]

### Final Answer:
The value of \( \frac{1}{x} + \frac{1}{y} \) is \( \frac{2a}{b} \).

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