Question

Let $z_1$ and $z_2$ be n^th roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form
[IIT Screening 2001; Karnataka 2002]

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

Answer 1

The correct option is D 4k

$1^{1/n}=cos\frac{2r\pi }{n}+isin\frac{2r\pi }{n}$
$Let{z}_1=cos\frac{2r\pi }{n}+isin\frac{2r\pi }{n}$
$and{z}_2=cos\frac{2r_2\pi }{n}+isin\frac{2r_2\pi }{n}.$
Then $\angle Z_{1}O{Z}_2=amp\left(\frac{z_1}{z_2}\right)=amp\left(z_1\right)-amp\left(z_2\right)$
$=\frac{2(r_1-r_2)\pi }{n}=\frac{\pi }{2}$ (Given)
$\therefore n=4(r_1-r_2)=4\times$ integer, so n is of the form 4 k.