Question

If $3$ complex numbers are in $A.P$. then they lie on a circle in the complex plane

• A. True
• B. False
• C. Ambiguous
• D. Data insufficient

Answer 1

The correct option is B False

Let the three numbers be
$z_1$ $z_2$ and $z_3$.
Since they are in A.P
$z_2-z_1=z_3-z_2$
Or
$2z_2=z_3+z_1$
Or
$z_2=\frac{z_3+z_1}{2}$
Now if $Re\left(z\right)=0$ or $Im\left(z\right)=0$ where $z$ defines all the three complex numbers.
Then we get co-linear numbers.
Hence if 3 complex numbers are in A.P then it is not necessarily true that they lie on a circle.