I was thinking about the following problem:
Let $G_1$ and $G_2$ be the images of the disc $\{z\in \mathbb C:|z+1|<1\}$ under the transformations $w=\frac{(1-i)z+2}{(1+i)z+2}$ and $w=\frac{(1+i)z+2}{(1-i)z+2}$ respectively. Then which of the following statement is correct?
(a) $G_1=\{w\in \mathbb C:Im(w)<0\}$ and $G_2=\{w\in \mathbb C:Im(w)>0\},$
(b) $G_1=\{w\in \mathbb C:Im(w)>0\}$ and $G_2=\{w\in \mathbb C:Im(w)<0\},$
(c) $G_1=\{w\in \mathbb C:|w|>2\}$ and $G_2=\{w\in \mathbb C:|w|<2\},$
(d) $G_1=\{w\in \mathbb C:|w|<2\}$ and $G_2=\{w\in \mathbb C:|w|>2\}.$
I was trying to express $z$ in terms of $w$ and then put the value in the relation $|z+1|<1$ and replace $w$ with $u+iv$ where $u$=$Re(w)$ and $v$=$Im(w)$. But the whole process in lengthy and also i could not reach desired result. Please help.Thanks in advance for your time.