Question

Circle A has a center at `(5 ,-4 )` and a radius of `6`. Circle B has a center at `(-4 ,-8 )` and a radius of `5`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

The circles overlap.

Explanation:

The distance between the center of A at `color(blue)(""(5,-4))`
and the center of B at `color(green)(""(-4,-8))`
is
`color(white)("XXX")d=sqrt((color(blue)(5)-(color(green)(-4)))^2+((color(blue)(-4))-(color(green)(-8)))^2)`

`color(white)("XXX")=sqrt(9^2+4^2)`

`color(white)("XXX")=sqrt(97)`

`color(white)("XXX")~~9.848858`

Circle A with a radius of `color(red)(6)`
covers `color(red)(6)` units of this distance
and
Circle B with a radius of `color(magenta)(5)`
covers `color(magenta)(5)` units of this distance

Between the two circles,
they cover `color(red)(6)+color(magenta)(5)=11` units.

Since the distance between them is only (approx.) `9.85`
the circles must overlap ( by approx. `11-9.85 = 1.15`

As verification, here is what the circles look like: