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.

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 $\frac{2}{3}$.
