maths...complexed numbers (transformation)

a)Show that the transformation

w=(z-1)/z
maps |z-1|=1 in the z-plane onto |w|=|w-1| in the w-plane.
ANSWER
w=(z-1)/z
wz=z-1
1=z-wz
1=z(1-w)
z=1/(1-w)
|z-1|=1 becomes
|1/(1-w)-1|=1
|[1-(1-w)]/(1-w)|=1
|w/(1-w)|=1
|w|=|w-1|
The region |z-1|=<1 in the z-plane is mapped onto the region T in the w-plane.
b) Shade the region T on an Argand diagram.
ANSWER
from (a)
|z-1|<=1 becomes
|w|<=|w-1|
let w=x+yi
x^2+y^2<=(x-1)^2+y^2
x^2+y^2<=x^2-2x+1+y^2
2x-1<=0
x<=1/2
So the region T in the w-plane is x<=1/2
你知點畫架啦﹐cheese !