Let `|z_(1)|=1, |z_(2)|=2, |z_(3)|=3 and z_(1)+z_(2)+z_(3)=3+sqrt5i`, then the value of `Re(z_(1)bar(z_(2))+z_(2)bar(z_(3))+z_(3)bar(z_(1)))` is equalto (where `z_(1), z_(2) and z_(3)` are complex numbers)

Options

A. 1
B. `-1`
C. `-(1)/(2)`
D. 0

Answer

To solve the problem, we need to find the value of \( \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \) given the conditions on the magnitudes of the complex numbers \( z_1, z_2, z_3 \) and their sum.

### Step 1: Understand the given conditions

We are given:
- \( |z_1| = 1 \)
- \( |z_2| = 2 \)
- \( |z_3| = 3 \)
- \( z_1 + z_2 + z_3 = 3 + \sqrt{5} i \)

### Step 2: Calculate the modulus of the sum

To find the modulus of the sum \( z_1 + z_2 + z_3 \):
\[ |z_1 + z_2 + z_3| = |3 + \sqrt{5} i| = \sqrt{3^2 + (\sqrt{5})^2} = \sqrt{9 + 5} = \sqrt{14} \]

### Step 3: Apply the triangle inequality

By the triangle inequality, we know:
\[ |z_1 + z_2 + z_3| \leq |z_1| + |z_2| + |z_3| \]
Substituting the known values:
\[ \sqrt{14} \leq 1 + 2 + 3 = 6 \]
This inequality holds, but we need to check if equality can occur.

### Step 4: Check for equality condition

Equality in the triangle inequality holds if \( z_1, z_2, z_3 \) are collinear in the complex plane. This means they can be represented as:
\[ z_1 = e^{i\theta}, \quad z_2 = 2e^{i\theta}, \quad z_3 = 3e^{i\theta} \]
for some angle \( \theta \).

### Step 5: Express the sum

If we assume they are collinear:
\[ z_1 + z_2 + z_3 = e^{i\theta} + 2e^{i\theta} + 3e^{i\theta} = (1 + 2 + 3)e^{i\theta} = 6e^{i\theta} \]
Setting this equal to \( 3 + \sqrt{5} i \) gives:
\[ 6e^{i\theta} = 3 + \sqrt{5} i \]

### Step 6: Find \( e^{i\theta} \)

Dividing both sides by 6:
\[ e^{i\theta} = \frac{1}{2} + \frac{\sqrt{5}}{6} i \]
Now, we can find the real and imaginary parts.

### Step 7: Calculate \( z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} \)

Using the expressions:
\[ z_1 = e^{i\theta}, \quad z_2 = 2e^{i\theta}, \quad z_3 = 3e^{i\theta} \]
We compute:
\[ z_1 \overline{z_2} = e^{i\theta} \cdot 2e^{-i\theta} = 2 \]
\[ z_2 \overline{z_3} = 2e^{i\theta} \cdot 3e^{-i\theta} = 6 \]
\[ z_3 \overline{z_1} = 3e^{i\theta} \cdot e^{-i\theta} = 3 \]
Adding these:
\[ z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1} = 2 + 6 + 3 = 11 \]

### Step 8: Find the real part

Since \( 11 \) is a real number, we have:
\[ \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 11 \]

### Final Answer

The value of \( \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \) is \( \boxed{11} \).