Complex variable.

Question

Let $z_{0},z_{1} \in \mathbb{C}$ and $c>0$. Show that $$ S = \left\lbrace z \in \mathbb{C} : \left|\left| \frac{z - z_{0}}{z - z_{1}} \right|\right| = c \right\rbrace$$ is a circumference if and only if $c \neq 1$. Also show that S is the circumference of radius $1$ if and only if $\overline{z_{0}} z_{1} = 1$.

I already did the first part of the question, I'm having trouble with the second part though. I have reduced $\left|\left| \frac{z - z_{0}}{z - z_{1}} \right|\right| = c$ in a way that I can see how the radius of the circumference behaves, yet I can't find the relationship between the fact that $\overline{z_{0}} z_{1} = 1$ with what I get. Thank you very much!

Edit: I'll add my calculations: $$\left|\left| \frac{z - z_{0}}{z - z_{1}} \right|\right|^2 = c^2$$ $$ \Rightarrow \frac{\left|\left| z - z_{0} \right|\right|^2}{\left|\left| z - z_{1} \right|\right|^2} = c^2$$ $$\Rightarrow \frac{||z||^2 - 2 \text{Re}(z \overline{z_{0}}) + ||z_{0}||^2}{||z||^2 - 2 \text{Re}(z \overline{z_{1}}) + ||z_{1}||^2} = c^2$$ $$x^2 + y^2 - 2x x_{0} - 2y y_{0} + x_{0}^2 + y_{0}^2 = c^2(x^2 + y^2 - 2x x_{1} - 2y y_{1} + x_{1}^2 + y_{1}^2)$$ $$(1-c^2)(x^2+y^2) + 2x(c^2 x_{1} - x_{0}) + 2y(c^2 y_{1} - y_{0}) = c^2(x_{1}^2 + y_{1}^2) - x_{0}^2 - y_{0}^2$$ $$ \Rightarrow x^2 + y^2 + 2 \Big(\frac{c^2 x_{1} - x_{0}}{1-c^2} \Big)x + 2 \Big(\frac{c^2 y_{1} - y_{0}}{1-c^2} \Big)y = \frac{c^2(x_{1}^2 + y_{1}^2) - x_{0}^2 - y_{0}^2}{1-c^2}$$ $$ \Rightarrow \Bigg(x + \frac{c^2 x_{1} - x_{0}}{1 - c^2} \Bigg)^2 + \Bigg(y + \frac{c^2 y_{1} - y_{0}}{1 - c^2} \Bigg)^2 = \frac{c^2(x_{1}^2 + y_{1}^2) - x_{0}^2 - y_{0}^2}{1-c^2} + \Big(\frac{c^2 x_{1} - x_{0}}{1-c^2} \Big)^2 + \Big(\frac{c^2 y_{1} - y_{0}}{1-c^2} \Big)^2$$ $$ \Rightarrow \Bigg(x + \frac{c^2 x_{1} - x_{0}}{1 - c^2} \Bigg)^2 + \Bigg(y + \frac{c^2 y_{1} - y_{0}}{1 - c^2} \Bigg)^2 = \frac{(1 - c^2)(c^2 x_{1}^2 + c^2 y_{1}^2 - x_{0}^2 - y_{0}^2) + c^4 x_{1}^2 - 2c^2 x_{1} x_{0} + x_{0}^2 + c^4 y_{1}^2 - 2c^2 y_{1} y_{0} + y_{0}^2}{(1 - c^2)^2}$$ $$ \Rightarrow \Bigg(x + \frac{c^2 x_{1} - x_{0}}{1 - c^2} \Bigg)^2 + \Bigg(y + \frac{c^2 y_{1} - y_{0}}{1 - c^2} \Bigg)^2 = \frac{c^2((x_{1} - x_{0})^2 + (y_{1} - y_{0})^2)}{(1 - c^2)^2}$$ $$ \therefore r = \frac{c ||z_{1} - z_{0}||}{|1 - c^2|}$$