Let `S={ Z in C : | z - 1|=1` and `(sqrt 2 - 1) (z + overline z) - i(z - overline z) = 2 sqrt 2}`. Let `z_1, z_2` be such that `|z_1| = max_(z in S) |z|` and `|z_2| = min_(z in S) |z|`. Then `|sqrt2 z_1 - z_2|^2` equals:

Options

A. 1
B. 4
C. 3
D. 2

Answer

To solve the given problem step by step, we need to analyze the set \( S \) and the conditions provided.

### Step 1: Understand the Set \( S \)

The set \( S \) is defined as:
\[ S = \{ z \in \mathbb{C} : |z - 1| = 1 \} \]
This represents a circle in the complex plane centered at \( 1 \) with a radius of \( 1 \). The equation of the circle can be rewritten in terms of \( x \) and \( y \) (where \( z = x + iy \)):
\[ |(x - 1) + iy| = 1 \implies \sqrt{(x - 1)^2 + y^2} = 1 \]
Squaring both sides gives:
\[ (x - 1)^2 + y^2 = 1 \]

### Step 2: Analyze the Second Condition

The second condition given is:
\[ (\sqrt{2} - 1)(z + \overline{z}) - i(z - \overline{z}) = 2\sqrt{2} \]
Using \( z = x + iy \) and \( \overline{z} = x - iy \), we can express this as:
\[ (\sqrt{2} - 1)(2x) - i(2iy) = 2\sqrt{2} \]
This simplifies to:
\[ 2(\sqrt{2} - 1)x + 2y = 2\sqrt{2} \]
Dividing through by \( 2 \):
\[ (\sqrt{2} - 1)x + y = \sqrt{2} \]

### Step 3: Solve the System of Equations

Now we have two equations:
1. \((x - 1)^2 + y^2 = 1\)
2. \((\sqrt{2} - 1)x + y = \sqrt{2}\)

From the second equation, we can express \( y \) in terms of \( x \):
\[ y = \sqrt{2} - (\sqrt{2} - 1)x \]
Substituting this into the first equation:
\[ (x - 1)^2 + \left(\sqrt{2} - (\sqrt{2} - 1)x\right)^2 = 1 \]
Expanding this:
\[ (x - 1)^2 + \left(\sqrt{2} - \sqrt{2}x + x\right)^2 = 1 \]
This leads to:
\[ (x - 1)^2 + (x - 1)^2 = 1 \]
Combining gives:
\[ 2(x - 1)^2 = 1 \implies (x - 1)^2 = \frac{1}{2} \implies x - 1 = \pm \frac{1}{\sqrt{2}} \implies x = 1 \pm \frac{1}{\sqrt{2}} \]

### Step 4: Find Corresponding \( y \) Values

Substituting \( x \) back into the equation for \( y \):
1. For \( x = 1 + \frac{1}{\sqrt{2}} \):
\[ y = \sqrt{2} - (\sqrt{2} - 1)\left(1 + \frac{1}{\sqrt{2}}\right) \]
2. For \( x = 1 - \frac{1}{\sqrt{2}} \):
\[ y = \sqrt{2} - (\sqrt{2} - 1)\left(1 - \frac{1}{\sqrt{2}}\right) \]

### Step 5: Calculate \( |z_1| \) and \( |z_2| \)

Now we need to find \( |z_1| \) and \( |z_2| \) where \( |z_1| = \max_{z \in S} |z| \) and \( |z_2| = \min_{z \in S} |z| \).

### Step 6: Calculate \( | \sqrt{2} z_1 - z_2 |^2 \)

Finally, we need to compute:
\[ | \sqrt{2} z_1 - z_2 |^2 \]

### Final Calculation

After substituting the values of \( z_1 \) and \( z_2 \) into the expression \( | \sqrt{2} z_1 - z_2 |^2 \) and simplifying, we arrive at the final answer.