I have to find conditions defining the following plane figures:
Where: $a=3$ and $b=7$
I know that circumference form is: $$\left |z-z_0 \right | =b$$
So, for c. with center $(3,3)$ and radius $7$ the equation would be something like: $$\left |z-3-3i \right | =7$$
But I don't know how to model the other points.
Can anyone help me?
for D, you have $\left(|z-a-ai| \le b\right) \cap \left(\Im(z) \ge a\right)$. This use of intersections allows you to composite several shapes together (in this case a circle and a half-plane) to create your shape. You'll need this for A as well, though I'll let you do that one.
For B, there's a neat trick that lets you pretend you're working in 2d vector space instead of the complex plane: given two complex numbers $u$ and $v$, you can find the dot product and 2d scalar analogue to 3d vector cross product of vectors $\vec u$ and $\vec v$ as $$\vec u \cdot \vec v = \Re\left(uv^*\right)$$ $$\vec u \times \vec v = \Im\left(uv^*\right)$$
Since a line (and thus, using an inequality, a half plane) can be expressed as an equation involving dot product, you can thus express B as such.