How can I find the number of complex numbers satisfying |z|=z+1+2i?

[utkarshakash]

Homework Statement

Find the number of complex numbers satisfying |z|=z+1+2i

Homework Equations

The Attempt at a Solution

Let z=x+iy
|x+iy| = (x+1)+i(2+y)
Squaring and taking modulus
|\sqrt{x^{2}+y^{2}}|^{2} = |(x+1)+i(2+y)|^{2}
x^{2}+y^{2} = (x+1)^{2}+(2+y)^{2}
Rearranging and simplifying I get
2x+4y+5=0

Now what to do next? Also Is there any other way to solve this question?

 

[HallsofIvy]

You seem to have missed an important point- the absolute value of a complex number is real. Since the left side of the equation is real, the right side must be. z must be of the form z= x- 2i so that z+ 1+ 2i= x+1. What you did was correct but how many points on the line 2x+ 4y+ 5= 0 also satisfy y= -2?