If the roots `alpha and beta` of the equation , `x^(2) - sqrt(2) x + c = 0 ` are complex for some real number `c ne 1` and `|(alpha - beta)/( 1 - alpha beta)|` = 1, then a value of c is :

Options

A. `- 2 + sqrt(6)`
B. `4 - sqrt(6)`
C. `-1 +sqrt(2)`
D. `- 1 + sqrt(6)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is:
\[ x^2 - \sqrt{2} x + c = 0 \]
From this, we can identify:
- \( a = 1 \)
- \( b = -\sqrt{2} \)
- \( c = c \)

### Step 2: Use the quadratic formula to find the roots
The roots of the equation can be found using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values:
\[ \alpha, \beta = \frac{\sqrt{2} \pm \sqrt{(\sqrt{2})^2 - 4 \cdot 1 \cdot c}}{2 \cdot 1} \]
\[ \alpha, \beta = \frac{\sqrt{2} \pm \sqrt{2 - 4c}}{2} \]

### Step 3: Determine the condition for complex roots
For the roots to be complex, the discriminant must be negative:
\[ 2 - 4c < 0 \]
This simplifies to:
\[ c > \frac{1}{2} \]

### Step 4: Calculate the expression \(|\alpha - \beta|\)
The difference between the roots \(|\alpha - \beta|\) can be calculated as:
\[ |\alpha - \beta| = \left| \frac{\sqrt{2} + \sqrt{2 - 4c}}{2} - \left( \frac{\sqrt{2} - \sqrt{2 - 4c}}{2} \right) \right| \]
This simplifies to:
\[ |\alpha - \beta| = \left| \frac{2\sqrt{2 - 4c}}{2} \right| = |\sqrt{2 - 4c}| \]

### Step 5: Calculate \(|1 - \alpha \beta|\)
The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = c \]
Thus:
\[ |1 - \alpha \beta| = |1 - c| \]

### Step 6: Set up the equation based on the given condition
We are given that:
\[ \left| \frac{\alpha - \beta}{1 - \alpha \beta} \right| = 1 \]
Substituting the expressions we found:
\[ \left| \frac{\sqrt{2 - 4c}}{1 - c} \right| = 1 \]

### Step 7: Solve the equation
This leads to two cases:
1. \( \sqrt{2 - 4c} = 1 - c \)
2. \( \sqrt{2 - 4c} = -(1 - c) \)

#### Case 1:
Squaring both sides:
\[ 2 - 4c = (1 - c)^2 \]
Expanding the right side:
\[ 2 - 4c = 1 - 2c + c^2 \]
Rearranging gives:
\[ c^2 + 2c - 1 = 0 \]
Using the quadratic formula:
\[ c = \frac{-2 \pm \sqrt{(2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2} \]

#### Case 2:
Squaring both sides:
\[ 2 - 4c = (1 - c)^2 \]
This case leads to the same equation as Case 1.

### Step 8: Check the values of \(c\)
We have two potential values:
1. \( c = -1 + \sqrt{2} \)
2. \( c = -1 - \sqrt{2} \)

Since \( c \neq 1 \) and we need \( c > \frac{1}{2} \), we check:
- \( -1 + \sqrt{2} \approx 0.414 \) (not valid)
- \( -1 - \sqrt{2} \) is negative (not valid)

Thus, the only valid value is:
\[ c = -1 + \sqrt{2} \]

### Final Answer:
The value of \( c \) is:
\[ c = -1 + \sqrt{2} \]