What will be the equation of an annulus?

Question

Let us consider a circle with radius 4 and a concentric circle with a radius 2. Now if the annular region is shaded, what will be the equation of the shaded annular region?

Answer 1

A circle with radius $r$ centered at $(a,b)$ has the equation $$(x-a)^2+(y-b)^2=r^2$$ (which is seen by using the Pythagorean theorem to calculate the distance between an arbitrary point $(x,y)$ on the circle and the center $(a,b)$).

Every point inside the shaded annulus is on a circle of radius $r$ for some $2 \leq r \leq 4$. Thus, the equation of the shaded annulus is give by: $$2^2 \leq (x-a)^2+(y-b)^2 \leq 4^4.$$

Answer 2

$$(x^2+y^2-4)(x^2+y^2-16)\le0.$$