Question

Consider an 'n' sided regular polygon with the origin as its centre. If $z_1$ be the complex number representing a vertex $a_1$ of the polygon, then the number associated with the vertex that is adjacent to $a_1$ is

• A.
• B.
• C.
• D.

Answer 1

Answer: (A), (D)

The correct options are
A

D

$z_2$ and $z_n$ are the vertex adjacent to $z_1$

We want to find the complex numbers $z_n$ and $z_2$. We will find them using concept of rotation .For what we need $\theta$. We know

$\theta$ = $\frac{2\pi }{n}$ [each angle is same, and total angle is 2$\pi$]

$\frac{z_n-0}{z_1-0}$ = $\left|\frac{z_n}{z_1}\right|$ $z_1$ $e^{i\frac{2\pi }{n}}$ ( Applying the concept of rotation)

|$z_n$| = |$z_1$|

$\Rightarrow$ $z_n$ = z $e^{i\frac{2\pi }{3}}$

Also $\frac{z_n-0}{z_1-0}$ = $\left|\frac{z_1}{z_2}\right|$ $\times$ $e^{i\frac{2\pi }{n}}$

$\Rightarrow$ $z_2$ = $z_1$ $e^{-i\frac{2\pi }{n}}$