If `k gt 0, k ne 1, and z_(1), z_(2) in C`, then `|(z-z_(1))/(z-z_(2))|` = k represents

Options

A. a circle
B. an ellipse
C. a parabola
D. a hyperbola

Answer

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

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

where \( k > 0 \) and \( k \neq 1 \), and \( z_1, z_2 \) are complex numbers.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( \left| \frac{z - z_1}{z - z_2} \right| = k \) indicates that the ratio of the distances from the point \( z \) to the points \( z_1 \) and \( z_2 \) is a constant \( k \).

2. **Rewriting in Terms of Distances**:
This can be rewritten as:
\[ |z - z_1| = k |z - z_2| \]
This means that the distance from \( z \) to \( z_1 \) is \( k \) times the distance from \( z \) to \( z_2 \).

3. **Geometric Interpretation**:
The above equation represents a circle in the complex plane. The points \( z \) that satisfy this equation are those that maintain a constant ratio of distances to the two fixed points \( z_1 \) and \( z_2 \).

4. **Finding the Center and Radius**:
To find the center and radius of this circle, we can manipulate the equation. We can express \( z \) in terms of its real and imaginary parts:
\[ z = x + iy, \quad z_1 = x_1 + iy_1, \quad z_2 = x_2 + iy_2 \]

5. **Squaring Both Sides**:
Squaring both sides, we have:
\[ |z - z_1|^2 = k^2 |z - z_2|^2 \]
Expanding both sides:
\[ (x - x_1)^2 + (y - y_1)^2 = k^2 \left( (x - x_2)^2 + (y - y_2)^2 \right) \]

6. **Expanding and Rearranging**:
Expanding both sides gives:
\[ (x^2 - 2xx_1 + x_1^2 + y^2 - 2yy_1 + y_1^2) = k^2 (x^2 - 2xx_2 + x_2^2 + y^2 - 2yy_2 + y_2^2) \]
Rearranging terms leads to:
\[ (1 - k^2)x^2 + (1 - k^2)y^2 - 2( x_1 - k^2 x_2)x - 2( y_1 - k^2 y_2)y + (x_1^2 + y_1^2 - k^2(x_2^2 + y_2^2)) = 0 \]

7. **Identifying the Circle**:
The equation can be rewritten in the standard form of a circle, confirming that it represents a circle in the complex plane.

### Conclusion:
Thus, the equation \( \left| \frac{z - z_1}{z - z_2} \right| = k \) represents a circle in the complex plane.