# Homework Help: Find no. of Complex Numbers

1. Oct 3, 2012

### utkarshakash

1. The problem statement, all variables and given/known data
Find the number of complex numbers satisfying $|z|=z+1+2i$

2. Relevant equations

3. 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?

2. Oct 3, 2012

### 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?