Let a be a complex number such that |a|<1 and z1,z2,......,zn be the vertices of a polygon such that zk=1+a+a2+......ak,then find equation of circle within which the vertices of the polygon lie.
\(z_k=\frac{1-a^{k+1}}{1-a}\)
\(\Rightarrow z_k-\frac{1}{1-a}=\frac{-a^{k+1}}{1-a}\)
\(\Rightarrow |z_k-\frac{1}{1-a}|=|\frac{-a^{k+1}}{1-a}|=\frac{|a^{k+1}|}{|1-a|}\)
\(\Rightarrow |z_k-\frac{1}{1-a}| < \frac{1}{|1-a|}\), since |a|<1
The circle, \(|z_k-\frac{1}{1-a}|=\frac{1}{|1-a|}\) is one possibility.
Thank you sir.