It seems complex numbers has just gotten a new standard of difficulty. I'am unable to post my working at the moment due to not having any start with the Maths Syntax used here. The question as a whole is connected in a manner that I can't solve the later parts getting done with the first part so please help me out.
The variable complex number z is given b:
z = 2cos@ + i(1-2sin@).
where @ takes all values in the interval pi < @ <= pi.
a, Show that lz-il = 2 for all values of @. Hence sketch, in an Argand Diagram, the locus of the point representing z.
b, Prove that the real part of 1/z+2-i is constant for -pi < @ < pi