If ` z_(1) and z_(2) ` are two unimodular complex numbers such that ` z_(1)^(2) + z_(2)^(2) = 3 " then " ( z_(1) + bar(z)_(1))^(2) + (z_(2) + bar(z)_(2))^(2)` =

Options

A. 10
B. 9
C. 6
D. 3

Answer

To solve the problem, we need to find the value of \( (z_1 + \bar{z_1})^2 + (z_2 + \bar{z_2})^2 \) given that \( z_1 \) and \( z_2 \) are unimodular complex numbers such that \( z_1^2 + z_2^2 = 3 \).

### Step-by-Step Solution:

1. **Understanding Unimodular Complex Numbers**:
A unimodular complex number \( z \) satisfies \( |z| = 1 \). This means that if \( z_1 = x_1 + iy_1 \), then \( x_1^2 + y_1^2 = 1 \). Similarly, for \( z_2 = x_2 + iy_2 \), we have \( x_2^2 + y_2^2 = 1 \).

2. **Expressing \( z_1^2 + z_2^2 \)**:
We can express \( z_1^2 \) and \( z_2^2 \) as follows:
\[ z_1^2 = (x_1 + iy_1)^2 = x_1^2 - y_1^2 + 2ix_1y_1 \]
\[ z_2^2 = (x_2 + iy_2)^2 = x_2^2 - y_2^2 + 2ix_2y_2 \]
Therefore,
\[ z_1^2 + z_2^2 = (x_1^2 - y_1^2 + x_2^2 - y_2^2) + i(2x_1y_1 + 2x_2y_2) \]
Given that \( z_1^2 + z_2^2 = 3 \), the imaginary part must be zero, leading to:
\[ 2x_1y_1 + 2x_2y_2 = 0 \]

3. **Finding \( (z_1 + \bar{z_1})^2 + (z_2 + \bar{z_2})^2 \)**:
We know that \( z_1 + \bar{z_1} = 2x_1 \) and \( z_2 + \bar{z_2} = 2x_2 \). Thus,
\[ (z_1 + \bar{z_1})^2 + (z_2 + \bar{z_2})^2 = (2x_1)^2 + (2x_2)^2 = 4x_1^2 + 4x_2^2 \]

4. **Using the Condition \( x_1^2 + y_1^2 = 1 \) and \( x_2^2 + y_2^2 = 1 \)**:
From the equations \( x_1^2 + y_1^2 = 1 \) and \( x_2^2 + y_2^2 = 1 \), we can express \( y_1^2 \) and \( y_2^2 \):
\[ y_1^2 = 1 - x_1^2 \quad \text{and} \quad y_2^2 = 1 - x_2^2 \]

5. **Substituting in the Condition**:
From the condition \( x_1^2 - y_1^2 + x_2^2 - y_2^2 = 3 \), we can substitute \( y_1^2 \) and \( y_2^2 \):
\[ x_1^2 - (1 - x_1^2) + x_2^2 - (1 - x_2^2) = 3 \]
Simplifying this gives:
\[ 2x_1^2 + 2x_2^2 - 2 = 3 \implies 2x_1^2 + 2x_2^2 = 5 \implies x_1^2 + x_2^2 = \frac{5}{2} \]

6. **Final Calculation**:
Now substituting back into our expression for \( (z_1 + \bar{z_1})^2 + (z_2 + \bar{z_2})^2 \):
\[ 4(x_1^2 + x_2^2) = 4 \cdot \frac{5}{2} = 10 \]

Thus, the final answer is:
\[ \boxed{10} \]