Sketching complex numbers in coordinate system

Question

Hi guys i want to sketch these set of complex numbers in coordinate system, i hope you can help me.

$a.\{z\in \mathbb{C}||z-1|+|z+1|<4\}$

$b.\{z\in \mathbb{C}| \mathrm{Im}((1-i)z)=0\}$

$c.\{z\in \mathbb{C}|1<|z+3i|<2\}$

Thanks in advance:)

Answer 1

Hint: For (b), $$\text{Im}((1-i)(x+iy))=\text{Im}((x+y)+i(y-x))=(y-x)$$ so what is the area in $\mathbb R^2$ when you are given $$y-x=0$$ For(c): If you set $x+iy=z$ then $|z+3i|=2\equiv|x+i(y+3)|=2~\equiv~\sqrt{x^2+(y+3)^2}=2$ shows a circle in $\mathbb R^2$ centered at $(0,-3)$ with radius $2$. The same is true for $|z+3i|=1$. Now what is $1<|z+3i|<2$? Isn't it the area between two circle which do not contain their borders?