How can can I find out if the two disks intersect in the plane?
I am not sure if I should answer such a question, but here we go:

Let the first disk be $D_1 = \langle R_1, (x_1, y_1)\rangle$ and the second disk be $D_2 = \langle R_2, (x_2, y_2)\rangle$, where $R_1$ and $R_2$ is the radius of the first and second disk respectively, as well as $(x_1, y_1)$ and $(x_2, y_2)$ is the center of the first and second disk respectively. Further, let $d$ be the distance of the two centers; that is, $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$ Then, the two disks intersect if $$d \leq R_1 + R_2$$ and do not intersect otherwise.

Note that if $d = R_1 + R_2$, then the two disks have only one point in common.