Question

Circle A has a center at `(8 ,7 )` and an area of `81 pi`. Circle B has a center at `(4 ,2 )` and an area of `36 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

The circles overlap.

Explanation:

The distance between the centers of Circle A and Circle B is
`color(white)("XXX")d=sqrt((8-4)^2+(7-2)^2) = sqrt(41)~~6.403`

`"Area"_A=pir_A^2=81pi`
`color(white)("XXX")rarr r_A^2=81`
`color(white)("XXX")rarr r_a=9`

`"Area"_B=pir_B^2=36pi`
`color(white)("XXX")rarr r_B^2=36`
`color(white)("XXX")rarr r_B=6`

Since the sum of the radii is greater than the distance between the centers, the circles overlap.

In fact along the line joining the centers of the circles, the circles overlap by
`color(white)("XXX")(9+6) - 6.403 = 8.597 " units"`