Complex numbers equation

[matemaprvic]

Hello!
I've got a problem with these equations:
|z| - iz = 1-2i

and

Im(z^2) = 4z-3

My book says that the solution for the first one is z=1-2i, but when I put that in the equation, it doesn't seem right. When I try to solve it, i get that x (or a) =2, and I dont know what to do with y (b).
And in the second one, I have no idea what I should do.
So, if anyone is willing to help me, I'd be very grateful.

 

[tkhunny]

|z| - iz = 1 - 2i

Well, let's rewrite it a little and think about it.

|z| = 1 - 2i + iz

This suggests 1 - 2i + iz must be REAL. Why?

A little Brute Force, using z = a+bi, the right hand side becomes 1 - 2i + i(a+bi) = 1 - 2i + ai - b = (1-b) + i(a-2)

Since that last form must also be REAL, we have a = 2.

This leaves the answer in your book wrong (assuming we have the right problem and the right answer). Where does that leave you as far as a complete solution?