Question

Circle A has a center at `(3 ,5 )` and a radius of `1`. Circle B has a center at `(-2 ,1 )` and a radius of `3`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

The circles do not overlap.
There is a minimum distance of (approximately) `2.4` between them.

Explanation:

The length of the line segment between A at `(3,5)` and B at `(-2,1)` is given by the Pythagorean Theorem as
`color(white)("XXX")L_(AB)=sqrt((3-(-2))^2+(5-1)^2)`
`color(white)("XXXXXX") = sqrt(41)`
`color(white)("XXXXXX")~~6.4`

The radius of A covers `1` unit of this length
and the radius of B covers another `3` units
for a total of `4` units covered by one or the other of the circles.

This leaves `6.4-4=2.4` units uncovered
and this is the minimum distance between the two circles.