IF `S = {z in C : bar(z) = iz^(2)}`, then the maximum value of `|z - sqrt(3) - i |^(2)` in S is ________

Answer

To solve the problem, we need to find the maximum value of \( |z - \sqrt{3} - i|^2 \) given the condition \( \bar{z} = iz^2 \).

### Step 1: Analyze the given condition
We start with the equation:
\[ \bar{z} = iz^2 \]
Taking the modulus on both sides, we have:
\[ |\bar{z}| = |iz^2| \]
Since \( |\bar{z}| = |z| \) and \( |i| = 1 \), we can write:
\[ |z| = |z|^2 \]

### Step 2: Solve for the modulus of \( z \)
From the equation \( |z| = |z|^2 \), we can factor it:
\[ |z|^2 - |z| = 0 \implies |z|(|z| - 1) = 0 \]
This gives us two possibilities:
1. \( |z| = 0 \) (which is not useful since it leads to \( z = 0 \))
2. \( |z| = 1 \)

Thus, we conclude that:
\[ |z| = 1 \]

### Step 3: Express \( z \) in polar form
Since \( |z| = 1 \), we can express \( z \) as:
\[ z = e^{i\theta} \quad \text{for some } \theta \in [0, 2\pi) \]

### Step 4: Calculate \( |z - \sqrt{3} - i|^2 \)
We need to find:
\[ |z - \sqrt{3} - i|^2 \]
Substituting \( z = e^{i\theta} \):
\[ |e^{i\theta} - \sqrt{3} - i|^2 \]
This can be rewritten as:
\[ |e^{i\theta} - (\sqrt{3} + i)|^2 \]

### Step 5: Find the maximum value of the expression
Let \( w = e^{i\theta} \). The expression becomes:
\[ |w - (\sqrt{3} + i)|^2 \]
This represents the square of the distance from the point \( w \) on the unit circle (centered at the origin) to the point \( (\sqrt{3}, 1) \).

### Step 6: Determine the distance from the center to the point
The distance from the origin to the point \( (\sqrt{3}, 1) \) is:
\[ d = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 7: Use the radius of the circle
The radius of the unit circle is \( 1 \). Therefore, the maximum distance from the point \( (\sqrt{3}, 1) \) to any point on the unit circle is:
\[ \text{Maximum distance} = d + \text{radius} = 2 + 1 = 3 \]

### Step 8: Square the maximum distance
Finally, we need the square of this maximum distance:
\[ |z - \sqrt{3} - i|^2 = 3^2 = 9 \]

### Conclusion
Thus, the maximum value of \( |z - \sqrt{3} - i|^2 \) in the set \( S \) is:
\[ \boxed{9} \]