(i) Show that any line in the complex plane $\mathbb{C}$ can be represented in the form

$$\bar{c} z+c \bar{z}+r=0,$$

where $c \in \mathbb{C}$ and $r \in \mathbb{R}$.

(ii) If $z$ and $u$ are two complex numbers for which

$$\left|\frac{z+u}{z+\bar{u}}\right|=1$$

show that either $z$ or $u$ is real.

(iii) Show that any Möbius transformation

$$w=\frac{a z+b}{c z+d} \quad(b c-a d \neq 0)$$

that maps the real axis $z=\bar{z}$ into the unit circle $|w|=1$ can be expressed in the form

$$w=\lambda \frac{z+k}{z+\bar{k}}$$

where $\lambda, k \in \mathbb{C}$ and $|\lambda|=1$.