According to what has been provided in Textbook: The translation of $\beta$ of $C$ is the circle centered at $z_0+\beta$ with the same radius. If the line L is $\{z:Re(Az+B) = 0\}$, then the translation of L by $\beta$ is the line $\{w: Re(Aw+B-A\beta) = 0\}$