Question

If $z_1$ and $z_2$ are the $nth$ roots of unity, then $arg\left(\frac{z_1}{z_2}\right)$ is a multiple of

• A. $n\pi$
• B. $\frac{3\pi }{n}$
• C. $\frac{2\pi }{n}$
• D. None of these

Answer 1

The correct option is C $\frac{2\pi }{n}$

Given, $z_1$ and $z_2$ are the $nth$ roots of unity
$\therefore z_1=e^{12\pi r/n}$ and $z_2=e^{i2\pi s/n}$
where, $r$ and $s$ are integers from $0$ to $n$.
$\therefore arg\left(\frac{z_1}{z_2}\right)=arg\left(z_1\right)-are\left(z_2\right)$
$=\frac{2r\pi }{n}-\frac{2s\pi }{n}$
$=2(r-s)\frac{\pi }{n}=k\left(\frac{2\pi }{n}\right)$
where, $k=r-s$
Here, $arg\left(\frac{z_1}{z_2}\right)$ is a multiple of $\frac{2\pi }{n}$.