If `z_(1),z_(2)` and `z_(3)` be unimodular complex numbers, then the maximum value of `|z_(1)-z_(2)|^(2)+|z_(2)-z_(3)|^(2)+|z_(3)-z_(1)|^(2)`, is

Options

A. 6
B. 9
C. 12
D. 3

Explanation

To find the maximum value of \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) for unimodular complex numbers \( z_1, z_2, z_3 \), we can follow these steps:

### Step 1: Understand Unimodular Numbers
Unimodular complex numbers are those that lie on the unit circle in the complex plane. This means that for any unimodular complex number \( z \), we have \( |z| = 1 \). Therefore, we can express \( z_1, z_2, z_3 \) as:
\[ z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2}, \quad z_3 = e^{i\theta_3} \]
for some angles \( \theta_1, \theta_2, \theta_3 \).

### Step 2: Expand the Expression
We want to calculate:
\[ y = |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \]
Using the property of modulus, we can expand each term:
\[ |z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2}) = |z_1|^2 + |z_2|^2 - z_1 \overline{z_2} - \overline{z_1} z_2 \]
Since \( |z_1| = |z_2| = 1 \), we have \( |z_1|^2 + |z_2|^2 = 2 \). Thus,
\[ |z_1 - z_2|^2 = 2 - z_1 \overline{z_2} - \overline{z_1} z_2 \]

### Step 3: Combine the Terms
Applying the same expansion for the other two terms:
\[ |z_2 - z_3|^2 = 2 - z_2 \overline{z_3} - \overline{z_2} z_3 \]
\[ |z_3 - z_1|^2 = 2 - z_3 \overline{z_1} - \overline{z_3} z_1 \]
Now, substituting these back into \( y \):
\[ y = (2 - z_1 \overline{z_2} - \overline{z_1} z_2) + (2 - z_2 \overline{z_3} - \overline{z_2} z_3) + (2 - z_3 \overline{z_1} - \overline{z_3} z_1) \]
\[ y = 6 - (z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3}) \]

### Step 4: Analyze the Minimum Value
The terms \( z_i \overline{z_j} \) can be expressed as \( e^{i(\theta_i - \theta_j)} \). The sum \( z_1 \overline{z_2} + z_2 \overline{z_1} + z_2 \overline{z_3} + z_3 \overline{z_2} + z_3 \overline{z_1} + z_1 \overline{z_3} \) can be minimized. The minimum value occurs when the angles are spaced evenly around the circle.

### Step 5: Maximum Value Calculation
The minimum value of the sum \( z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} + \text{conjugates} \) is \( -3 \) when the angles are \( 0, \frac{2\pi}{3}, \frac{4\pi}{3} \).

Thus, substituting back into \( y \):
\[ y = 6 - (-3) = 9 \]

### Conclusion
The maximum value of \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) is \( \boxed{9} \).