Question

Let $z_1$ and $z_2$ be the $n^{th}$ roots of unity, which are ends of a line segment that subtends a right angle at the origin. Then, $n$ must be of the form

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

Answer 1

Answer: (D)

$4k$

The $n^{th}$ roots of unity lie on a circle, where the angle between any 2 consecutive roots is $\frac{2\pi }{n}$.
Hence, $\frac{2r\pi }{n}=\frac{\pi }{2}$
$\Rightarrow 4r=n$
Hence, option D is correct.