The locus of the center of a circle which touches the circles `|z-z_1|=a, |z-z_2=b|` externally will be

Options

A. an ellipse
B. a hyperbola
C. a circle
D. none of these

Answer

To find the locus of the center of a circle that touches the circles defined by the equations \( |z - z_1| = a \) and \( |z - z_2| = b \) externally, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Circles**:
- The first circle has center \( z_1 \) and radius \( a \).
- The second circle has center \( z_2 \) and radius \( b \).

2. **Define the Circle that Touches Externally**:
- Let the center of the circle that touches both circles externally be \( z \) and its radius be \( r \).

3. **Set Up the Distance Equations**:
- Since the circle with center \( z \) touches the first circle externally, we have:
\[ |z - z_1| = a + r \]
- Since it touches the second circle externally, we have:
\[ |z - z_2| = b + r \]

4. **Express \( r \) in Terms of \( z \)**:
- From the first equation, we can express \( r \):
\[ r = |z - z_1| - a \]
- Substitute this expression for \( r \) into the second equation:
\[ |z - z_2| = b + (|z - z_1| - a) \]
- Rearranging gives:
\[ |z - z_2| - |z - z_1| = b - a \]

5. **Recognize the Hyperbola Property**:
- The equation \( |z - z_2| - |z - z_1| = b - a \) represents a hyperbola. This is because it describes the set of points \( z \) such that the difference in distances from two fixed points (the centers of the circles) is constant.

6. **Conclusion**:
- Therefore, the locus of the center \( z \) of the circle that touches both given circles externally is a hyperbola.