If `z_1,z_2,z_3` are complex number , such that `|z_1|=2, |z_2|=3, |z_3|=4` , the maximum value `|z_1-z_2|^(2) + |z_2-z_3|^2 + |z_3-z_1|^2` is :

Options

A. 58
B. 29
C. 87
D. None of these

Answer

To find the maximum value of the expression \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) given the magnitudes of the complex numbers \( |z_1| = 2 \), \( |z_2| = 3 \), and \( |z_3| = 4 \), we can follow these steps:

### Step 1: Write the expression in terms of magnitudes
We start by expanding the expression:

\[ |z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2 \text{Re}(z_1 \overline{z_2}) \]
\[ |z_2 - z_3|^2 = |z_2|^2 + |z_3|^2 - 2 \text{Re}(z_2 \overline{z_3}) \]
\[ |z_3 - z_1|^2 = |z_3|^2 + |z_1|^2 - 2 \text{Re}(z_3 \overline{z_1}) \]

### Step 2: Combine the terms
Now, we combine these three expansions:

\[ |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 = (|z_1|^2 + |z_2|^2 + |z_3|^2) + (|z_2|^2 + |z_3|^2 + |z_1|^2) - 2(\text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1})) \]

### Step 3: Substitute the magnitudes
Substituting the given magnitudes:

\[ |z_1|^2 = 4, \quad |z_2|^2 = 9, \quad |z_3|^2 = 16 \]

Thus, we have:

\[ |z_1|^2 + |z_2|^2 + |z_3|^2 = 4 + 9 + 16 = 29 \]

### Step 4: Write the expression
Now, substituting back into the expression:

\[ |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 = 2 \times 29 - 2(\text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1})) \]

### Step 5: Maximize the expression
To maximize this expression, we need to minimize the term \( \text{Re}(z_1 \overline{z_2}) + \text{Re}(z_2 \overline{z_3}) + \text{Re}(z_3 \overline{z_1}) \). The minimum value of this term occurs when the angles between the vectors representing \( z_1, z_2, z_3 \) are such that they are as far apart as possible.

### Step 6: Use the triangle inequality
Using the triangle inequality, we can find the maximum distance:

\[ |z_1 - z_2| \leq |z_1| + |z_2| = 2 + 3 = 5 \]
\[ |z_2 - z_3| \leq |z_2| + |z_3| = 3 + 4 = 7 \]
\[ |z_3 - z_1| \leq |z_3| + |z_1| = 4 + 2 = 6 \]

### Step 7: Calculate the maximum value
Thus, the maximum values of the squares are:

\[ |z_1 - z_2|^2 \leq 5^2 = 25 \]
\[ |z_2 - z_3|^2 \leq 7^2 = 49 \]
\[ |z_3 - z_1|^2 \leq 6^2 = 36 \]

Adding these maximum values:

\[ |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \leq 25 + 49 + 36 = 110 \]

### Conclusion
Thus, the maximum value of \( |z_1 - z_2|^2 + |z_2 - z_3|^2 + |z_3 - z_1|^2 \) is \( \boxed{110} \).