The denominator of `( a + sqrt( a^(2) -b^(2) ) )/( a- sqrt(a^(2) -b^(2) ) ) + ( a - sqrt( a^(2) - b^(2) ) )/( a + sqrt(a^(2)- b^(2) ) )` is ________.

Options

A. `a^2`
B. `b^2`
C. `a^2 -b^2`
D. `(4a^2 - 2b^2)/( b)`

Answer

To find the denominator of the expression

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

we will follow these steps:

### Step 1: Identify the Denominators
The denominators of the two fractions are:
1. \( a - \sqrt{a^2 - b^2} \)
2. \( a + \sqrt{a^2 - b^2} \)

### Step 2: Find the Least Common Multiple (LCM)
To combine the two fractions, we need to find the LCM of the denominators. The LCM of \( a - \sqrt{a^2 - b^2} \) and \( a + \sqrt{a^2 - b^2} \) is simply their product:

\[ \text{LCM} = (a - \sqrt{a^2 - b^2})(a + \sqrt{a^2 - b^2}). \]

### Step 3: Apply the Difference of Squares
Using the difference of squares formula, we can simplify the product:

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

### Conclusion
Thus, the denominator of the given expression is:

\[ b^2. \]