Question

If $z$ and $\overline{z}$ represent adjacent vertices of a regular polygon of $n$ sides with centre at origin and if $\frac{Imz}{Rez}=\sqrt{2}-1$, then the value of $n$ is equal to

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

The correct option is D 8

Now
Let $z_1$ be the first vertex of the polygon in the first quadrant,
Hence
$arg\left(z_1\right)=\frac{2\pi }{8}\times \frac{1}{2}=\frac{\pi }{8}$. ...
(angle between two corresponding vertices will be $\frac{2\pi }{n}$, hence angle made by one vertex with the horizontal x-axis will be exactly half of it).

$\therefore \frac{Im\left(z\right)}{Re\left(z\right)}$ $=tan\left(arg\right(z\left)\right)$

$=tan\left(\frac{\pi }{n}\right)$

$=\sqrt{2}-1$

Hence, $\frac{\pi }{n}=ta{n}^{-1}(\sqrt{2}-1)$

$\Rightarrow \frac{\pi }{n}=\frac{\pi }{8}$

Thus $n=8$.

Hence, the option 'D' is correct.