u(t)=αt+β is a straight line in
C.
z(t)=u(t)1 is an inversion of
u(t) combined with a reflection relative to the horizontal axis (a.k.a. complex conjugation). It is known that an inversion of a straight line not passing through 0 is a circle passing through 0 (
https://en.wikipedia.org/wiki/Inversive_geometry#Inversion_in_a_circle).
To prove more formally can do this variable transformation:
αt+β1=α1t+γ1, where
γ=αβ. Since multiplication by
α1 is just scaling and rotation it is enough to prove that
z(t)=t+γ1 represents a circle. The center of such circle is easy to guess:
c=2bi1, so what remains to show is that
∣∣∣t+bi1−2bi1∣∣∣=2b1