If a complex number z lie on a circle of radius `(1)/(2)` units, then the complex number `omega=-1 +4z` will always lie on a circle of radius k units, where k is equal to

Answer

To solve the problem, we need to analyze the transformation of the complex number \( z \) that lies on a circle of radius \( \frac{1}{2} \) to the new complex number \( \omega = -1 + 4z \).

### Step-by-Step Solution:

1. **Understanding the Circle for z**:
Since \( z \) lies on a circle of radius \( \frac{1}{2} \), we can express this mathematically as:
\[ |z| = \frac{1}{2} \]

2. **Expressing omega in terms of z**:
We have the transformation given by:
\[ \omega = -1 + 4z \]

3. **Rearranging the equation**:
We can rearrange the equation to express \( z \) in terms of \( \omega \):
\[ \omega + 1 = 4z \]
Therefore, we can write:
\[ z = \frac{\omega + 1}{4} \]

4. **Finding the modulus**:
We know that \( |z| = \frac{1}{2} \). We can substitute \( z \) in terms of \( \omega \):
\[ \left| \frac{\omega + 1}{4} \right| = \frac{1}{2} \]

5. **Simplifying the modulus**:
To simplify this, we multiply both sides by 4:
\[ |\omega + 1| = 2 \]

6. **Interpreting the result**:
The equation \( |\omega + 1| = 2 \) represents a circle in the complex plane centered at \( -1 \) with a radius of \( 2 \).

7. **Conclusion**:
Therefore, the radius \( k \) of the circle on which \( \omega \) lies is:
\[ k = 2 \]

### Final Answer:
The value of \( k \) is \( 2 \).