Question

lf $z_1,z_2$ are $n^{th}$ roots of unity which are ends of a line segment that subtends $\frac{\pi }{2}$ at the origin.

then $n$ is of the form.

• A. $4k+1$
• B. $4k+2$
• C. $4k+3$
• D. $4k$

Answer 1

The correct option is D $4k$

$z_1=e^{i\frac{2k_1\pi }{n}}$ $z_2=e^{i\frac{2k_2\pi }{n}}$

Given, $z_1=z_2{e}^{i{}^{\pi }{/}_2}$

$\Rightarrow e^{i\left(\frac{2k_1-2k_2}{n}\right)\pi }=e^{i{}^{\pi }{/}_2}$
$\Rightarrow \frac{2(k_1-k_2)\pi }{n}=\frac{\pi }{2}$
$or2=4(k_1-k_2)=4k$