Let `|z_(1)|=3, |z_(2)|=2 and z_(1)+z_(2)+z_(3)=3+4i`. If the real part of `(z_(1)bar(z_(2))+z_(2)bar(z_(3))+z_(3)bar(z_(1)))` is equal to 4, then `|z_(3)|` is equal to (where, `i^(2)=-1`)

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will use the information given about the complex numbers \( z_1, z_2, \) and \( z_3 \).

### Step 1: Understand the given information
We know:
- \( |z_1| = 3 \)
- \( |z_2| = 2 \)
- \( z_1 + z_2 + z_3 = 3 + 4i \)

### Step 2: Calculate the modulus of \( z_1 + z_2 + z_3 \)
To find the modulus of \( z_1 + z_2 + z_3 \):
\[ |z_1 + z_2 + z_3| = |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Use the property of modulus
We can use the property of modulus:
\[ |z_1 + z_2 + z_3|^2 = |z_1|^2 + |z_2|^2 + |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]
Substituting the known values:
\[ 5^2 = |z_1|^2 + |z_2|^2 + |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]
\[ 25 = 3^2 + 2^2 + |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]

### Step 4: Substitute the values of \( |z_1|^2 \) and \( |z_2|^2 \)
Calculating \( |z_1|^2 \) and \( |z_2|^2 \):
\[ |z_1|^2 = 9, \quad |z_2|^2 = 4 \]
Substituting these values:
\[ 25 = 9 + 4 + |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]
\[ 25 = 13 + |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 25 - 13 \]
\[ |z_3|^2 + 2 \cdot \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 12 \]

### Step 6: Substitute the value of the real part
We know that the real part \( \text{Re}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) = 4 \):
\[ |z_3|^2 + 2 \cdot 4 = 12 \]
\[ |z_3|^2 + 8 = 12 \]

### Step 7: Solve for \( |z_3|^2 \)
Subtracting 8 from both sides:
\[ |z_3|^2 = 12 - 8 = 4 \]

### Step 8: Find \( |z_3| \)
Taking the square root:
\[ |z_3| = \sqrt{4} = 2 \]

Thus, the modulus of \( z_3 \) is \( |z_3| = 2 \).