Question

If the complex numbers \( z_1, z_2, z_3 \) are in AP, then they lie on a:

• A. circle
• B. parabola
• C. line
• D. ellipse

Hint:

When complex numbers are in arithmetic progression, their corresponding points in the complex plane are always collinear.

Answer 1

The Correct Option is C

Step 1: Understanding the given condition.
The complex numbers \( z_1, z_2, z_3 \) are in arithmetic progression (AP). For three complex numbers to be in AP, the middle term \( z_2 \) must be the arithmetic mean of \( z_1 \) and \( z_3 \). This can be expressed as: \[ z_2 = \frac{z_1 + z_3}{2}. \] This condition implies that the points corresponding to \( z_1, z_2, z_3 \) form a straight line in the complex plane.
Step 2: Geometrical interpretation.
When complex numbers are in AP, the points represented by these numbers are collinear. This means that the points lie on a straight line in the complex plane.
Step 3: Conclusion.
Therefore, if the complex numbers \( z_1, z_2, z_3 \) are in AP, they lie on a line, and the correct answer is (c).