Let a be a complex number such that `|a| lt 1` and `z_(1),z_(2)…..` be vertices of a polygon such that `z_(k)=1+a+a^(2)+a^(3)+a^(k-1)`.
Then, the vertices of the polygon lie within a circle.

Options

A. `|z-a|=a`
B. `|z-1/(1-a)|=|1-a|`
C. `|z-1/(1-a)|=1/(|1-a|)`
D. `|z-(1-a)|=|1-a|`

Answer

C

Explanation

We have,
`z_(k)=1+a+a^(2)+…..+a^(k-1)=(1-a^(k))/(1-a)`
`rArr =1/(1-a)=a^(k)/(1-a)`
`rArr |z_(k)-1/(1-a)|=(|a|)^(k)/(|1-a|)= lt 1/(|1-a|)`
`rArr` lies within the circle `=|z-1/(1-a)|=1/(|1-a|)`