SOLUTION: Hello, I need help with the following math problem: Find all complex numbers {{{ z }}} such that (the link to the equation is below): http://latex.artofproblemsolving.com/c/

|z|^2 - 2*conj(z) + i*z = 2i.   (1)   (conj(z) means z-conjugated . . . )

Let z = x + yi,  where "x" and "y" are unknown real numbers.

Then 

|z|^2 =    (real number);

2*conj(z) = 2x - 2yi;

i*z = -y +x*i             (why?  Try to understand it on your own. It is elementary . . . )

Now substitute all this stuff into (1). You will get

 = .   (2)

It is an equality of complex numbers. It means that their real parts are equal and imaginary parts are equal.
In other words, you have these two equations, first for real parts and the second for imaginary parts

 = 0,        (3)       ( which represents a circle . . . ) and
2y + x = 2.                (4)        ( which represents a straight line )

Now you need to solve this system and to find "x" and "y"  (the intersection points).


I don't want to deprive you, Nicole, the pleasure to do it on your own.

Can you complete it?

If not, let me know, I will help you in it.
In this case, send me a message through the "Thank you" window, and do not forget to mention 
the ID number of the problem  ( 1046469 ) in order for I could identify it.

Good luck!