Question

Circle A has a radius of `2` and a center at `(3 ,1 )`. Circle B has a radius of `4` and a center at `(8 ,3 )`. If circle B is translated by `<-4 ,-1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

They don't overlap. The minimum distance is `2-sqrt(2)`.

Explanation:

The new center of circle B is `(4,2)`.

The distance between the centers of the two circles is

`sqrt((4-3)^2 + (2-1)^2)= sqrt(2)`

Circle B is the larger circle and all points on the circle are 4 units from its center.

The radius of circle A is 2 and the center of A is `sqrt(2)` units from the center of Circle B so the farthest any point on A can be from the center of B is `2+sqrt(2)`. Since `2+sqrt(2)<4` no point on A overlaps any point on B.

The minimum distance between points on A and B is actually the radius of B, which is 4, minus the farthest any point on A can be from the center of B, which is `2+sqrt(2)`:

`4-(2+sqrt(2)) = 2-sqrt(2)`