If ` 2x = sqrt(a) - (1)/(sqrt(a))` , then the value of ` (sqrt(x^(2) + 1))/(x + sqrt(x^(2) +1))` is

Options

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

Answer

To solve the equation \( 2x = \sqrt{a} - \frac{1}{\sqrt{a}} \) and find the value of \( \frac{\sqrt{x^2 + 1}}{x + \sqrt{x^2 + 1}} \), we will follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ 2x = \sqrt{a} - \frac{1}{\sqrt{a}} \]
We square both sides:
\[ (2x)^2 = \left(\sqrt{a} - \frac{1}{\sqrt{a}}\right)^2 \]
This gives us:
\[ 4x^2 = \left(\sqrt{a}\right)^2 - 2\left(\sqrt{a}\right)\left(\frac{1}{\sqrt{a}}\right) + \left(\frac{1}{\sqrt{a}}\right)^2 \]

### Step 2: Simplify the right side
Now, simplify the right side:
\[ 4x^2 = a - 2 + \frac{1}{a} \]
Thus, we can rewrite it as:
\[ 4x^2 = a + \frac{1}{a} - 2 \]

### Step 3: Express \( x^2 \)
Now, we can express \( x^2 \):
\[ x^2 = \frac{a + \frac{1}{a} - 2}{4} \]

### Step 4: Find \( x^2 + 1 \)
Next, we need to find \( x^2 + 1 \):
\[ x^2 + 1 = \frac{a + \frac{1}{a} - 2}{4} + 1 \]
This simplifies to:
\[ x^2 + 1 = \frac{a + \frac{1}{a} - 2 + 4}{4} = \frac{a + \frac{1}{a} + 2}{4} \]

### Step 5: Calculate \( \sqrt{x^2 + 1} \)
Now we find \( \sqrt{x^2 + 1} \):
\[ \sqrt{x^2 + 1} = \sqrt{\frac{a + \frac{1}{a} + 2}{4}} = \frac{\sqrt{a + \frac{1}{a} + 2}}{2} \]

### Step 6: Substitute into the expression
Now substitute \( \sqrt{x^2 + 1} \) back into the expression:
\[ \frac{\sqrt{x^2 + 1}}{x + \sqrt{x^2 + 1}} = \frac{\frac{\sqrt{a + \frac{1}{a} + 2}}{2}}{x + \frac{\sqrt{a + \frac{1}{a} + 2}}{2}} \]

### Step 7: Simplify the denominator
To simplify the denominator, we express \( x \) in terms of \( a \):
\[ x = \frac{\sqrt{a} - \frac{1}{\sqrt{a}}}{2} \]
Substituting this into the denominator:
\[ x + \sqrt{x^2 + 1} = \frac{\sqrt{a} - \frac{1}{\sqrt{a}}}{2} + \frac{\sqrt{a + \frac{1}{a} + 2}}{2} \]
This can be simplified further.

### Step 8: Final simplification
After simplification, we find that:
\[ \frac{\sqrt{x^2 + 1}}{x + \sqrt{x^2 + 1}} = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \frac{\sqrt{x^2 + 1}}{x + \sqrt{x^2 + 1}} \) is:
\[ \frac{1}{2} \]