Consider plane R2 as a complex plane. Consider the transformation $z -> \frac{z+1}{z-1}$. Which circles under this transformation will become lines?
I'm trying to understand how these transformations work and currently am extremely confused about how to work something like this. Any tips
A line in $w$-plane ($w\in\Bbb{C}$) is given by the equation $$(a+bi)w+(a-bi)\overline{w}=c$$ where $a,b,c\in\Bbb{R}$, $a^2+b^2\neq 0$. Let $w=f(z)=\frac{z+1}{z-1}$. Then the pre-image of the line above under $f(z)$ is $$(2a-c)x^2+(2a-c)y^2+2by+2cx-2a-c=0$$ which is a circle when $2a-c\neq 0$ and a line otherwise. It can be checked that it is passing through $1=(1,0)$.
Conversely, any circle or line passing through $1$ in $z$-plane is sent to a line because $f(1)=\infty.$ Or we can arrange the coefficients. For example, the circle $x^2+y^2-2x_0x-2y_0y+2x_0-1=0$ passing through $1$ in $z$-plane is sent to the line $(\frac{x_0-1}{2}+y_0i)w+(\frac{x_0-1}{2}-y_0i)\overline{w}=x_0$ in $w$-plane.