Circle A has a center at #(5 ,4 )# and an area of #4 pi#. Circle B has a center at #(2 ,8 )# and an area of #9 pi#. Do the circles overlap? If not, what is the shortest distance between them?

1 Answer
Nov 29, 2016

The circles touch, but they do not overlap. The shortest distance between them is #0#.

Explanation:

The distance between the points #(5,4)# and #(2,8)# is
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#
#d=sqrt((2-5)^2+(8-4)^2)#
#d=sqrt((-3)^2+4^2)#
#d=sqrt(9+16)#
#d=sqrt(25)=5#

If the circles overlap, the sum of their radii will be greater than #d=5.# If they don't, the shortest distance between them will be what we get when we subtract both radii from #5#.

Let #r_1# and #r_2# be the radius of circles 1 and 2, respectively. Then:

#A_1=pir_1^2#
#=>r_1=sqrt(A_1/pi)=sqrt((4cancelpi)/cancelpi)=sqrt(4)=2#

Similarly,
#r_2=sqrt(A_2/pi)=sqrt((9cancelpi)/cancelpi)=sqrt(9)=3#

We get
#"     "r_1+r_2#
#=2+3#
#=5"           "=d#

Since the sum of the radii matches the distance between the two circles' centers, the circles touch, but they do not overlap. The shortest distance between the two circles is #0#.