Graphing a set of Complex Numbers I came across the following question:
Make a graph of the following set:
$$E=\{z \in \mathbb{C}\big| |z+i|=2|z|\} $$
But I have no clue how to find elements from this set. I looked at some more trivial numbers, such as $0$,$1$ and $i$ but non seem to be in this set. Also simplifying the condition doesn't give me any clue.
Can someone please help me with understanding this set? 
 A: I'd rewrite the condition as $\left|\frac{z-i}{z}\right|=2$, then get rid of tge modulus: 
$$
\frac{z-i}{z}=2e^{it}
$$
Now solve for $z$.  Then plug in various $t$ and see what $z$ you get.
A: The complex numbers in question correspond to certain points in $R^2$ as shown below

So, the question can be formulated as a problem in coordinate geometry:
Picture the curve described by the following equation:
$$\sqrt{(y+1)^2+x^2}=2\sqrt{(x^2+y^2)}$$
or
$$(y+1)^2+x^2=4(x^2+y^2).$$
We get then
$$3y^2-2y+3x^2-1=0$$
which is the equation of a circle of radius $\frac23$ centered at $\left(0,\frac13\right)$. 
Indeed. Divide both sides of the equation above by $3$ and rearrange again:
$$y^2-\frac23y+x^2-\frac13=0.$$
Taking the complete square of the $y$ term we get
$$\left(y-\frac13\right)^2+x^2=\frac49.$$
A: When $\text{z}\in\mathbb{C}$:


*

*$$\left|\text{z}+i\right|=\left|\left(\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right)+i\right|=\left|\Re\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)i\right|=\sqrt{\Re^2\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)^2}$$

*$$2\left|\text{z}\right|=2\left|\Re\left[\text{z}\right]+\Im\left[\text{z}\right]i\right|=2\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$


So, when set those two equal:
$$\sqrt{\Re^2\left[\text{z}\right]+\left(1+\Im\left[\text{z}\right]\right)^2}=2\sqrt{\Re^2\left[\text{z}\right]+\Im^2\left[\text{z}\right]}$$
