Question

Circle A has a center at `(5 ,3 )` and an area of `4 pi`. Circle B has a center at `(1 ,2 )` and an area of `16 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

The circles overlap.

Explanation:

Since `"Area"_"circle"=pir^2`
`color(white)("XXX")`Circle A (with area `4pi`) has a radius `r_A=2`
and
`color(white)("XXX")`Circle B (with area `16pi`) has a radius of `r_B=4`

The length of the line segment from the center of A at `(5,3)` to the center of B at `(1,2)` can be calculated using the Pythagorean Theorem as
`color(white)("XXX")d_"AB" = sqrt( (5-1)^2+(3-2)^2)=sqrt(17)` (approximately `4.123`)

Circle A covers `2` units along this line segment and
Circle B covers `4` units along this line segment.

Since `2+4 > d_"AB"`
`color(white)("XXX")`the circles overlap (by `(2+4)-4.123 = 1.877` units)