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?

1 Answer
Dec 27, 2016

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#