If `|z|=k` and `omega=(z-k)/(z+k)`, then Re`(omega)`=

Options

A. 0
B. k
C. `1/k`
D. `-1/k`

Explanation

To solve the problem, we need to find the real part of the complex number \( \omega \) defined as:

\[ \omega = \frac{z - k}{z + k} \]

given that \( |z| = k \).

### Step-by-step Solution:

1. **Understanding the Modulus Condition**:
We are given that \( |z| = k \). This means that \( z \) lies on a circle of radius \( k \) in the complex plane. We can express this condition mathematically as:
\[ |z|^2 = k^2 \]
which implies:
\[ z \cdot \overline{z} = k^2 \]

2. **Expressing \( \overline{z} \)**:
From the modulus condition, we can express \( \overline{z} \) in terms of \( z \):
\[ \overline{z} = \frac{k^2}{z} \]

3. **Substituting into \( \omega \)**:
Now, we substitute \( z \) and \( \overline{z} \) into the expression for \( \omega \):
\[ \omega = \frac{z - k}{z + k} \]

4. **Finding the Real Part of \( \omega \)**:
To find the real part of \( \omega \), we can use the property of complex numbers. The real part of a complex number \( w \) can be expressed as:
\[ \text{Re}(\omega) = \frac{1}{2} \left( \omega + \overline{\omega} \right) \]

First, we need to find \( \overline{\omega} \):
\[ \overline{\omega} = \frac{\overline{z} - k}{\overline{z} + k} \]

Substituting \( \overline{z} = \frac{k^2}{z} \):
\[ \overline{\omega} = \frac{\frac{k^2}{z} - k}{\frac{k^2}{z} + k} \]

5. **Simplifying \( \omega + \overline{\omega} \)**:
Now we calculate \( \omega + \overline{\omega} \):
\[ \omega + \overline{\omega} = \frac{z - k}{z + k} + \frac{\frac{k^2}{z} - k}{\frac{k^2}{z} + k} \]

To simplify this, we can find a common denominator and combine the fractions. However, we notice that the two terms are symmetric and will ultimately lead to cancellation.

6. **Conclusion**:
After performing the algebra, we find that:
\[ \text{Re}(\omega) = 0 \]

### Final Answer:
Thus, the real part of \( \omega \) is:
\[ \text{Re}(\omega) = 0 \]