Find the image of the domain

1. The problem statement, all variables and given/known data

Find the image of the domain

[tex]\{z:x<0, -x+y<t\}[/tex] where t is a fixed positive integer, under the transformation

[tex]M(z)=\frac{z-1}{1-zi}[/tex]







3. The attempt at a solution

Im attempting to map the two boundary lines in the domain:

Since [tex]-i\mapsto\infty[/tex] the line [tex]x=0[/tex] (i.e the imaginary axis) is mapped to a line. To find this line I calculated two points of the mapping:

[tex]i\mapsto \frac{-1+i}{2}\:\:\:\:\:\:\:\:0\mapsto -1[/tex] which gives the equation of the line in cartesian coordinates as [tex]y=x+1.[/tex]



I know the the other boundary line [tex]y=t+x[/tex] is mapped to a circle. Here is my problem: I need to find the coordinates of any 3 points that are mapped to by 3 points on this line and then find the circle that has these three points on its boundary. I have been able to do this for t=1 but am having a hard time with the general form in terms of t.