Question

In the complex plane, let $z_1=\sqrt{3}+i$ and $z_2=\sqrt{3}-i$ be two adjacent vertices of an $n$-sided regular polygon centred at the origin. Then $n$ equals.

• A. $4$
• B. $6$
• C. $8$
• D. $12$

Answer 1

The correct option is B $6$

Clearly we observe that after plotting both the points on the graph

$\tan \theta =\frac{1}{\sqrt{3}}$

$\Rightarrow \theta ={30}^{\circ }$

Angle subtended by adjacent sides at the centre is $2\theta ={60}^{\circ }$

Now we know that sum of angles in a polygon subtended by each side at the centre is ${360}^{\circ }$.

But in an $n$ sided regular polygon, if each angle subtended is $\alpha$.

Then sum of the angles subtended is $n\alpha$.
Here $\alpha ={60}^{\circ }$

Hence, $n\alpha ={360}^{\circ }$

$\Rightarrow n=\frac{{360}^{\circ }}{{60}^{\circ }}=6$