Question

Circle A has a center at `(1 ,5 )` and an area of `12 pi`. Circle B has a center at `(8 ,1 )` and an area of `3 pi`. Do the circles overlap? If not, what is the shortest distance between them?

Answer 1

We first get the radii of the two circles.

Explanation:

Since `A=pir^2->r=sqrt(A/pi)`
Circle A: `r_A=sqrt((12cancelpi)/cancelpi)=sqrt12=2sqrt3`
Circle B: `r_B=sqrt((3cancelpi)/cancelpi)=sqrt3`

Then the distance between the centers:

`D^2=(Deltax)^2+(Deltay)^2` (Pythagoras)
`D^2=(8-1)^2+(1-5)^2=7^2+4^2=49+16=65`
`->D=sqrt65~~8.06`

Together:
The radii add up to `2sqrt3+sqrt3=3sqrt3=sqrt27~~5.20`

This is much smaller than the distance, so they do not overlap. The smallest distance between them is `sqrt65-sqrt27~~2.87`