Question

Let, $z_1$ and $z_2$ be $n$th roots of unity which subtend a right angle at the origin. Then n must be of the from

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

Answer 1

The correct option is D $4k$

$z_1$ and $z_2$ subtend right angle at the origin.

Then

$arg\frac{z_1}{z_2}=\frac{\pi }{2}$

$\therefore \frac{z_1}{z_2}=\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}0+i=i$

${\left(\frac{z_1}{z_2}\right)}^n=i^n=1$ ( nth root of unity )

$\therefore n=4k$ (as $i^4=1$)