Two circles have the following equations #(x -1 )^2+(y -4 )^2= 36 # and #(x +5 )^2+(y +5 )^2= 81 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

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Answer 1 · 2017-06-13T13:09:34

The circles overlap.

The radius of circle #A# is

#r_A=sqrt(36)=6#

The radius of circle #B# is

#r_B=sqrt(81)=9#

The distance between the center #=(1,4)# of circle #A# and the center #(-5,-5)# of circle #B# is

#d=sqrt((1+5)^2+(4+5)^2)#

#=sqrt(6^2+9^2)#

#sqrt(36+81)=sqrt117#

#=10.81#

The sum of the radii is

#r_A+r_B=6+9=15#

Therefore,

As #(r_A+r_B )> d#

the circles overlap.
graph{((x-1)^2+(y-4)^2-36)((x+5)^2+(y+5)^2-81)=0 [-28.87, 28.88, -14.43, 14.43]}