Question

If the complex numbers $z_1,z_2,z_3$are in an AP, then they lie on:

• A. a circle
• B. a parabola
• C. a straight line
• D. an ellipse

Answer 1

The correct option is C

a straight line

Explanation for the correct option.

By definition of the arithmetic progression we know that if $a,b,c$are in A.P then, $2b=a+c$.

So, if $z_1,z_2,z_3$ are in A.P then,

$\begin{array}{rcl}2z_2& =& z_1+z_3\\ z_2& =& \frac{z_1+z_3}{2}\end{array}$

It means that $z_2$ is the mid point of $z_1,z_3$ .

Therefore, $z_1,z_2,z_3$ lie on a straight line.

Hence, option C is correct.