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

Answer

A

Explanation

Given,
`z_(k) = 1 + a+ a^(2) +......+a^(k-1) = (1-a^(k))/(1-a)`
`rArr a_(k) -(1)/(1-a) = - (a^(k))/(1-a)`
`rArr |z_(k)-(1)/(1-a)|= (|a|^(k))/(|1-a|) lt (1)/(|1-a|)" " [because|a| lt 1]`
Hence , `z_(k)` lies within the circle.
`therefore |z-(1)/(1-a)|= (1)/(|1-a|)`