Q. Let `z_1` and `z_2` be nth roots of unity which subtend a right angle at the origin, then n must be the form `4k`.

Options

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

Answer

D

Explanation

The `n^(th)` roots of unity are given by
`e^(i(2rpi)/n), r=0,1,2,……,(n-1)`
So let
`z_(1)=e^(i2mpi)/n` and `z_(2)=e^(i2ppi)/n`, where `0 le m, p lt n, m ne p`
It is given that the line segment joining the points having affixes `z_(1)` and `z_(2)` subtends a right angle at the origin.
`therefore "arg"(z_(1)/z_(2))=+-pi/2`
`rArr (2mpi)/n-(2ppi)/n=+-pi/2`
`rArr n=+-4(m-p) rArr n=4k`, where k=m-p.