If `|(z-z_1)/(z-z_2)|=1`, where `z_1 and z_2` are fixed complex numbers and z is a variable 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 and z_2` as its interior points
D. circle with `z_1 and z_2` as its exterior points

Answer

To solve the problem, we start with the equation given:

\[ \left| \frac{z - z_1}{z - z_2} \right| = 1 \]

### Step 1: Understand the Meaning of the Modulus
The modulus of a complex number represents its distance from the origin in the complex plane. The equation states that the distance from \( z \) to \( z_1 \) is equal to the distance from \( z \) to \( z_2 \).

### Step 2: Rewrite the Equation
Since the modulus is equal to 1, we can rewrite the equation as:

\[ |z - z_1| = |z - z_2| \]

This means that the distance from \( z \) to \( z_1 \) is equal to the distance from \( z \) to \( z_2 \).

### Step 3: Geometric Interpretation
The condition \( |z - z_1| = |z - z_2| \) geometrically represents the set of points \( z \) that are equidistant from the points \( z_1 \) and \( z_2 \). This set of points is the perpendicular bisector of the line segment joining \( z_1 \) and \( z_2 \).

### Step 4: Conclusion
Thus, we conclude that the variable complex number \( z \) lies on the perpendicular bisector of the line segment joining the fixed complex numbers \( z_1 \) and \( z_2 \).

### Final Answer
The complex number \( z \) lies on the perpendicular bisector of the line segment joining \( z_1 \) and \( z_2 \).

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