can a complex math equation can create multiple closed area?

Question

example : equation of circle can easily represent case A.

can some single complex mathematical equation can create case B ?

Answer 1

Yes, this is possible. Suppose one of the circles is represented as the equation $f(x,y)=0$ and the other as $g(x,y)=0$. Then the picture of both circles would be represented by the equation $$\color{blue}{f(x,y)g(x,y)=0}.$$

(This applies also to drawing the union of two general curves, not just circles.)

For example, if one circle is $x^2+y^2=1$ and another is $(x-1)^2+(y-2)^2=4$, then plotting the equation $$\left(x^2+y^2-1\right)\left((x-1)^2+(y-2)^2-4\right)=0$$ will get you the picture of both circles.

Answer 2

Yes, two curves can be represented with a single equation joining them as their product.Right hand sides of the individual equations should be zero before multiplying.

Multiple curves can be drawn on same x-y plane. We can visualize them together or separately. Draw graphs of two or three curves separately on transparent plastic sheets. Superimpose the sheets along their respective axes coinciding them onto a common origin.

The single product equation you get is valid representation for what all curves in the set you see.