Graphical representation of complex numbers

Question

How would you graphically represent the following complex functions on the Argand plane (ie. not polar coordinates): $$|2i-z|=|z+1+3i|$$ and $$Re[(3-4i)z]>0$$ I know that $|z-a|=c$ is a circle centred at a with radius c and $Re(z)>0$ is the right hand side of the plane, but how do I use these to graph the above functions?

Answer 1

$|2i-z|$ represents the distance of $z$ from $2i$ and $|z+1+3i|$ represents the distance of $z$ from $-1-3i$. If the distance is the same then draw a line connecting these $2$ points. And then in the middle of this line draw another line perpendicular to it. This line represents your set.

For $z = x+yi$, $(3-4i)z = 3x+3yi-4xi + 4y$.

So $Re((3-4i)z) > 0 \iff 3x+4y>0 \iff y>-\frac{3x}{4}$

Answer 2

HINT:

For $|2i-z|=|z+1+3i|$: Distance from $(0,2)$ = Distance from $(-1,-3)$ in $\mathbb{C}$.

For $Re[(3-4i)z]>0$: If you visualize the geometric notion of multiplication of $\mathbb{C}$ numbers and try to keep the product have a positive $X$, you will realize that $z$ lies in the half plane just missing the origin and farthest from the point $(-3,4)$. This approach works because of the linear nature of the equation. For complicated stuff it is easier to rely on the algebraic methods i.e. substituting $z=x+iy$. Anyhow you can always be sure that this kind of inequality will always result in a (may be non-existent) region.