Suppose z and `omega` are two complex number such that Which of the following is true for z and `omega`?

Options

A. `Re(z) = Re(omega)=(1)/(2)`
B. `Im(z) = Im(omega)`
C. `Re(z) = Im(omega)`
D. `Im(z) =Re(omega)`

Explanation

To solve the problem involving the complex numbers \( z \) and \( \omega \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Define the Complex Numbers**:
Let \( z = x + i y \) and \( \omega = \alpha + i \beta \), where \( x, y, \alpha, \beta \) are real numbers.

2. **Square the Complex Numbers**:
We calculate \( z^2 \) and \( \omega^2 \):
\[ z^2 = (x + i y)^2 = x^2 - y^2 + 2xyi \]
\[ \omega^2 = (\alpha + i \beta)^2 = \alpha^2 - \beta^2 + 2\alpha \beta i \]

3. **Set Up the Equation**:
According to the problem, we have:
\[ z^2 + \omega^2 + i(\omega \bar{z} - \bar{\omega} z) = 4 \]
Here, \( \bar{z} = x - i y \) and \( \bar{\omega} = \alpha - i \beta \).

4. **Separate Real and Imaginary Parts**:
From the equation, we can separate the real and imaginary parts:
- Real part: \( x^2 - y^2 + \alpha^2 - \beta^2 = 4 \)
- Imaginary part: \( 2\alpha \beta + 2xy = 0 \)

5. **Analyze the Imaginary Part**:
From the imaginary part, we can express:
\[ \alpha \beta + xy = 0 \quad \Rightarrow \quad \alpha \beta = -xy \]

6. **Consider Cases**:
- **Case 1**: If \( \alpha = 0 \), then \( \beta \) can be \( \pm 1 \) (since \( \alpha^2 + \beta^2 = 1 \)).
- **Case 2**: If \( z = 0 \), then \( y = 0 \) and we find \( \alpha \) must also be \( 0 \).

7. **Conclusion**:
From the analysis, we find that if \( \alpha = 0 \), then \( \beta = \pm 1 \) leads us to \( \omega = \pm i \). Thus, the imaginary part of \( z \) is equal to the real part of \( \omega \).

### Final Answer:
The correct conclusion is that the imaginary part of \( z \) is equal to the real part of \( \omega \).