For {$z: |z + 3 - 3i| = 2$} determine:
a) the minimum possible value of $Im(z)$
For this I have drawn a graph by transforming {$z: |z + 3 - 3i| = 2$} into its Cartesian equation form, i.e $(x+3)^2 + (y-3)^2 = 4$. Produces a circle centre $(-3+3i)$ with radius $ = 2$
From the graph, the minimum possible value of $Im(z)$ is $1$.
b) the maximum possible value of $|Re(z)|$
Using the graph, the max possible value of $|Re(z)|$ is $5$.
I do not understand how to find the following though, but I am thinking maybe something to do with the Cartesian equation? Or is there a simpler way to do it?
c) the minimum possible value of $|z|$
d) the maximum possible value of $|z|$
e) the maximum possible value of $|\bar{z}|$ ($|z|$ conjugate)
Because $|z|=|\bar{z}|,$ it suffices to work with $|z|.$
Absolute value $|z|$ is the distance between $z$ and $0$ in the Argand plane.
As you have found correctly, the set of points $z$ such that $|z+3-3i|=2$ is a circle.
The points on the circle that are nearest/farthest to $0$ lie on the line passing through $0$ and the center $(-3+3i).$ Here, the line is angle bissector of the second quadrant.
The distance from $0$ to $(-3+3i)$ is $3\sqrt2.$
To obtain $\min z,$ subtract $2,$ which is the radius. Similarly, for $\max z$ add $2.$ $$\min z =3\sqrt 2 -2, \max z =3\sqrt 2+2.$$
If you prefer to continue with the equation $(x+3)^2+(y-3)^2=2,$ substitute $y=-x,$ (equation of the second bissector).