If `2x = sqrta + (1)/( sqrta) , a gt 0` then the volume of `(sqrt (x ^(2) -1 ))/( x - sqrt (x ^(2) -1))`

Options

A. `1/2 (a- 1)`
B. `a -1`
C. `a +1`
D. `1/2 (a +1)`

Answer

To solve the given problem, we start with the equation provided:

**Given:**
\[ 2x = \sqrt{a} + \frac{1}{\sqrt{a}}, \quad a > 0 \]

### Step 1: Solve for \( x \)
To find \( x \), we divide both sides of the equation by 2:
\[ x = \frac{1}{2} \left( \sqrt{a} + \frac{1}{\sqrt{a}} \right) \]

### Step 2: Find \( x^2 \)
Now, we square \( x \):
\[ x^2 = \left( \frac{1}{2} \left( \sqrt{a} + \frac{1}{\sqrt{a}} \right) \right)^2 \]
\[ x^2 = \frac{1}{4} \left( \sqrt{a} + \frac{1}{\sqrt{a}} \right)^2 \]
Expanding the square:
\[ x^2 = \frac{1}{4} \left( a + 2 + \frac{1}{a} \right) \]
\[ x^2 = \frac{a + 2 + \frac{1}{a}}{4} \]

### Step 3: Calculate \( x^2 - 1 \)
Now, we calculate \( x^2 - 1 \):
\[ x^2 - 1 = \frac{a + 2 + \frac{1}{a}}{4} - 1 \]
To combine the terms, we convert 1 to a fraction:
\[ x^2 - 1 = \frac{a + 2 + \frac{1}{a}}{4} - \frac{4}{4} \]
\[ x^2 - 1 = \frac{a + 2 + \frac{1}{a} - 4}{4} \]
\[ x^2 - 1 = \frac{a - 2 + \frac{1}{a}}{4} \]

### Step 4: Calculate \( \sqrt{x^2 - 1} \)
Now we find \( \sqrt{x^2 - 1} \):
\[ \sqrt{x^2 - 1} = \sqrt{\frac{a - 2 + \frac{1}{a}}{4}} = \frac{\sqrt{a - 2 + \frac{1}{a}}}{2} \]

### Step 5: Calculate \( x - \sqrt{x^2 - 1} \)
Next, we calculate \( x - \sqrt{x^2 - 1} \):
\[ x - \sqrt{x^2 - 1} = \frac{1}{2} \left( \sqrt{a} + \frac{1}{\sqrt{a}} \right) - \frac{\sqrt{a - 2 + \frac{1}{a}}}{2} \]
\[ = \frac{1}{2} \left( \sqrt{a} + \frac{1}{\sqrt{a}} - \sqrt{a - 2 + \frac{1}{a}} \right) \]

### Step 6: Calculate the volume expression
Finally, we need to find the expression:
\[ \frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} \]
Substituting our previous results:
\[ \frac{\frac{\sqrt{a - 2 + \frac{1}{a}}}{2}}{\frac{1}{2} \left( \sqrt{a} + \frac{1}{\sqrt{a}} - \sqrt{a - 2 + \frac{1}{a}} \right)} \]
This simplifies to:
\[ \frac{\sqrt{a - 2 + \frac{1}{a}}}{\sqrt{a} + \frac{1}{\sqrt{a}} - \sqrt{a - 2 + \frac{1}{a}}} \]

### Final Result
Thus, the volume of the given expression is:
\[ \frac{\sqrt{a - 2 + \frac{1}{a}}}{\sqrt{a} + \frac{1}{\sqrt{a}} - \sqrt{a - 2 + \frac{1}{a}}} \]