Let `a` be a complex number such that `|a|<1a n dz_1, z_2,z_3,... ` be the vertices of a polygon such that `z_k=1+a+a^2+...+a^(k-1)` for all `k=1,2,3, T h e nz_1, z_2` lie within the circle (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/(a+1)|=|a+1|`

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|)`