Question

The centre of a regular polygon of n sides is located at the point z=0 and one of its vertex $z_1$ is known. If $z_1$ be the vertex adjacent to $z_1$, then $z_2$ is equal to

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

Let A be the vertex with affix $z_1$.There are two possibilities of $z_2$, i.e., $z_2$ can be obtained by rotating $z_1$ through $\frac{2\pi }{n}$ either in clockwise or in anticlockwise direction.

$\therefore \frac{z_2}{z_1}$ = $\left|\frac{z_2}{z_1}\right|e^{\pm \frac{i2\pi }{n}}\Rightarrow z_2$ $z_1{e}^{\pm \frac{i2\pi }{n}}$, $(\because |z_2|$=$|z_1|)$

$\Rightarrow z_2$ = $z_1(cos\frac{2\pi }{n}\pm isin\frac{2\pi }{n})$