If `|(z-z_(1))//(z-z_(2))| = 3`, where `z_(1)` and `z_(2)` are fixed complex numbers and z is a variable complex complex number, then z lies on a

Options

A. circle with `z_(1)` as its interior point
B. circle with `z_(2)` as its interior point
C. circle with `z_(1)` as its exterior point
D. circle with `z_(2)` as its exterior point

Explanation

To solve the problem, we need to analyze the given condition:

Given:
\[ \left| \frac{z - z_1}{z - z_2} \right| = 3 \]

where \( z_1 \) and \( z_2 \) are fixed complex numbers and \( z \) is a variable complex number.

### Step-by-Step Solution:

1. **Understanding the Modulus Condition**:
The expression \(\left| \frac{z - z_1}{z - z_2} \right| = 3\) implies that the distance from \( z \) to \( z_1 \) is three times the distance from \( z \) to \( z_2 \).

2. **Rearranging the Equation**:
We can rewrite the equation as:
\[ |z - z_1| = 3 |z - z_2| \]

3. **Geometric Interpretation**:
The equation \( |z - z_1| = k |z - z_2| \) (where \( k = 3 \)) describes a circle in the complex plane. Specifically, it indicates that the point \( z \) lies on a circle whose center and radius depend on the points \( z_1 \) and \( z_2 \).

4. **Finding the Circle's Properties**:
- The center of the circle lies on the line segment joining \( z_1 \) and \( z_2 \).
- The ratio of the distances \( |z - z_1| \) and \( |z - z_2| \) defines the radius of the circle.

5. **Using the Internal and External Bisectors**:
The internal and external bisectors of the segment joining \( z_1 \) and \( z_2 \) can be used to find the points where the circle intersects the line joining \( z_1 \) and \( z_2 \).

6. **Conclusion**:
The locus of points \( z \) satisfying the given condition is a circle with \( z_1 \) and \( z_2 \) as points related to the circle's geometry.

### Final Answer:
Thus, the point \( z \) lies on a circle with the center determined by the fixed points \( z_1 \) and \( z_2 \).

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