How to sketch the region on the complex plane? I am going through a basic course on complex analysis. I have a problem in understanding the following.
E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ I want to know if this set is connected, closed, bounded but I do not know how to sketch this set or visualize it.
 A: I would write $z=x+\mathrm i y$ and then see what equation I got.
\begin{eqnarray*}
|z+\mathrm i| &=& 2|z| \\ \\
|(x+\mathrm iy)+\mathrm i| &=& 2|x+\mathrm iy| \\ \\
|x+(1+y)\mathrm i|&=& 2|x+\mathrm iy| \\ \\
\sqrt{x^2+(1+y)^2} &=& 2\sqrt{x^2+y^2} \\ \\
x^2+(1+y)^2 &=& 4(x^2+y^2) \\ \\
x^2+1+2y+y^2 &=& 4x^2+4y^2 \\ \\
1 &=& 3x^2 + 3y^2 -2y \\ \\
\frac{1}{3} &=& x^2 + y^2 - \frac{2}{3}y \\ \\
\frac{1}{3} &=& x^2 + \left(y - \frac{1}{3}\right)^2 - \frac{1}{9} \\ \\
\frac{4}{9} &=& x^2 + \left(y - \frac{1}{3}\right)^2
\end{eqnarray*}
You have a circle, centre $(0,\frac{1}{3})$ with radius $\frac{2}{3}$.
A: How to visualize:
$|z|$ can be thought of as the distance from the origin to the point $z.$
$|z-(a+bi)|$ would be the distance $z$ is from the point $a+bi.$
So here you have   
$\{z:|z+i| = 2|z|\}$ All $z$ whose distance from the origin is 1/2 their distance from the point $-i.$  Sounds like some sort of ellipse.
A little more analysis on this problem.  
$|z+i| = 2|z|$
if $z = x+iy, |z| = \sqrt{x^2+y^2}$
squaring both sides:
$x^2 + (y+1)^2 = 4 x^2 + 4y^2$
That is a circle.
$x^2 + (y-1/3)^2 = \frac 49$
