Circle A has a center at #(1 ,8 )# and an area of #18 pi#. Circle B has a center at #(8 ,1 )# and an area of #45 pi#. Do the circles overlap?

1 Answer
Nov 21, 2016

The circles overlap

Explanation:

Circle A
#area =pir_A^2=18pi#

So, #r_A=sqrt18=3sqrt2#

Circle B
#area=pir_B^2=45pi#

So, #r_B=sqrt45=3sqrt5#

The distance between the centers of the circles #A (x_A,y_A)# and #B(x_B,y_B)#

#d=sqrt((x_B-x_A)^2+(y_b-y_A)^2)#

#d=sqrt((8-1)^2+(1-8)^2)=sqrt(7^2+7^2)=7sqrt2=9.9#

The sum of the radii #=r_A+r_B=3sqrt2+3sqrt5=10.95#

Therefore, #d<=r_A+r_B#
So, the circles overlap

The equations of the circles are

#(x-1)^2+(y-8)^2=18#

#(x-8)^2+(y-1)^2 =45#
graph{((x-1)^2+(y-8)^2-18)((x-8)^2+(y-1)^2-45)=0 [-22.45, 23.15, -5.73, 17.09]}