Want to know if there is a better way to go about this problem.
Something nags at me that expanding the squares after squaring both sides is unnecessarily tedious.
ok I don't understand why the first method is false, but you can do the same thing |z - 2/(1+i)| = 2sqrt(2) -> circle of radius 2sqrt(2) centered around 1-i
but I don't know why the first one doesn't work
You have to be kidding
|y-2| = r is the circle of center 2 and radius r.
Ok so I found the mistake in
the set of points described by y is indeed the circle of center 2 and radius 4
to obtain z though, you have to take this set and multiply it by (1-i)/2 (since z = (1-i)y/2)
so this will rotate the circle around the origin (including the center).
In the end, you indeed find the center (1-i)*2/2 = (1-i) and the radius 2sqrt(2)