another solution to question 16c:
xz + yw has to be an obtuse rotation away from z which is a region spanned by the complex numbers iz and -z. in x and y, this is somewhere in the third quadrant which would be easier to show with a diagram but i cbb just take it as fact.
let a and b be positive real numbers.
the first boundary of this region is that zx+yw = -az where a is positive real --> x + y*w/z = -a
Taking imaginary parts, y*Im(w/z) = 0 --> y = 0
Taking real parts x + y*Re(w/z) = -a --> x = -a, meaning that x < 0 since a is positive real.
Hence one of the boundaries of the region is the line y = 0, x < 0 or the x axis on the left of the cartesian plane'
Another way to do this step is looking at zx+yw = -az, z and w are linearly independent vectors so you can instantly go to x = -a and w = 0.
the second boundary of the region is zx+yw = -biz where b is positive real --> x + y w/z = -bi
Taking real parts --> x + y*Re(w/z) = 0 --> y = -x/Re(w/z) --> |y| = |-x/Re(w/z)| --> -y = -x/|Re(w/z)| --> y = x/|Re(w/z)| where x < 0, y < 0
The complete solution is the region bounded by S = { (x,y) \in R^2 : y < 0, x < 0, y > x/|Re(w/z)|}