Line reflections on the complex plane

Question

Let $l$ be the line indicated below. For certain complex numbers $a$ and $b$, the function $g(z)=a\bar{z} + b$ represents a reflection across $l$. What is the value of $b$?

I've tried converting the line into a Cartesian graph (by taking note of the asymptotes), and I've done so successfully: the graph would be $y = x-2$. However, I'm ultimately unsure of how this could be useful.

Answer 1

Looking at the picture you have $g(0)=2-2i$ for example, which yields $$a\cdot\overline{0}+b=g(0)=2-2i.$$

Answer 2

The equation of the line seems to be $y=x+2.$ Reflection across this line is $$(x_1,y_1) \mapsto (y_1-2,x_1+2)$$ Identifying the point $(x,y)$ with the complex number $x+yi$ gives the equation $$y_1-2+(x_1+2)i=i(x_1-y_1i)+(-2+2i)$$ Thus $a=i,b=-2+2i$