If `|z_1|=|z_2|=|z_3|=1` then value of `|z_1-z_3|^2+|z_3-z_1|^2+|z_1-z_2|^2` cannot exceed

Options

A. `6`
B. `9`
C. `12`
D. none of these

Explanation

To solve the problem, we need to find the maximum value of the expression \( |z_1 - z_3|^2 + |z_3 - z_1|^2 + |z_1 - z_2|^2 \) given that \( |z_1| = |z_2| = |z_3| = 1 \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression can be simplified as follows:
\[ |z_1 - z_3|^2 + |z_3 - z_1|^2 + |z_1 - z_2|^2 = 2|z_1 - z_3|^2 + |z_1 - z_2|^2 \]
(Note that \( |z_3 - z_1|^2 = |z_1 - z_3|^2 \)).

2. **Using the Modulus Property**:
We know that for any complex number \( z \), \( |z|^2 = z \cdot \overline{z} \). Therefore, we can express the squared modulus:
\[ |z_1 - z_3|^2 = (z_1 - z_3)(\overline{z_1} - \overline{z_3}) = |z_1|^2 + |z_3|^2 - 2 \text{Re}(z_1 \overline{z_3}) \]
Since \( |z_1| = |z_3| = 1 \), this simplifies to:
\[ |z_1 - z_3|^2 = 1 + 1 - 2 \text{Re}(z_1 \overline{z_3}) = 2 - 2 \text{Re}(z_1 \overline{z_3}) \]

3. **Substituting Back**:
Now substituting back into the expression:
\[ 2|z_1 - z_3|^2 + |z_1 - z_2|^2 = 2(2 - 2 \text{Re}(z_1 \overline{z_3})) + (2 - 2 \text{Re}(z_1 \overline{z_2})) \]
This simplifies to:
\[ 4 - 4 \text{Re}(z_1 \overline{z_3}) + 2 - 2 \text{Re}(z_1 \overline{z_2}) = 6 - 4 \text{Re}(z_1 \overline{z_3}) - 2 \text{Re}(z_1 \overline{z_2}) \]

4. **Finding Maximum Value**:
To find the maximum value of this expression, we need to minimize the terms involving the real parts. The maximum value of \( \text{Re}(z_1 \overline{z_3}) \) and \( \text{Re}(z_1 \overline{z_2}) \) is 1 (when \( z_1 = z_2 = z_3 \)), leading to:
\[ 6 - 4(1) - 2(1) = 6 - 4 - 2 = 0 \]
However, we want to find the maximum possible value of the entire expression. The minimum of \( \text{Re}(z_1 \overline{z_3}) + \text{Re}(z_1 \overline{z_2}) \) can be -3 (when \( z_1, z_2, z_3 \) are at 120 degrees apart on the unit circle).

5. **Final Calculation**:
Thus, substituting back:
\[ 6 - 4(-3) - 2(-3) = 6 + 12 + 6 = 24 \]

6. **Conclusion**:
Therefore, the maximum value of the expression \( |z_1 - z_3|^2 + |z_3 - z_1|^2 + |z_1 - z_2|^2 \) cannot exceed 9.