If `z_(1),z_(2),z_(3)………….z_(n)` are in `G.P` with first term as unity such that `z_(1)+z_(2)+z_(3)+…+z_(n)=0`. Now if `z_(1),z_(2),z_(3)……..z_(n)` represents the vertices of `n`-polygon, then the distance between incentre and circumcentre of the polygon is

Options

A. `0`
B. `|z_(1)|`
C. `2|z_(1)|`
D. none of these

Answer

A

Explanation

`(a)` Let vertices be `1, alpha,alpha^(2),……….,alpha^(n-1)`
Given `1+alpha+alpha^(2)+…….+alpha^(n-1)=0impliesalpha^(n)-1=0`
`impliesz_(1),z_(2),z_(3),……….,z_(n)` are roots of `alpha^(n)=1`
which form regular polygon. So distance is zero.