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

• A. 4k+1
• B. 4k+2
• C. 4k+3
• D. 4k

Answer 1

The correct option is D 4k

$1^{\frac{1}{n}}=cos\frac{2r\pi }{n}+isin\frac{2r\pi }{n}Let{z}_1=cos\frac{2r_1\pi }{n}+isin\frac{2r_1\pi }{n}and{z}_1=cos\frac{2r_1\pi }{n}+isin\frac{2r_1\pi }{n}Then\angle Z_{1}O{Z}_2=amp\left(\frac{z_1}{z_2}\right)=amp\left(z_1\right)-amp\left(z_2\right)=\frac{2(r_1-r_2)\pi }{n}=\frac{\pi }{2}\left(Given\right)\therefore n=4(r_1-r_2)=4\times integer,sonisoftheform4k$.