Two circles have the following equations: #(x +6 )^2+(y -5 )^2= 64 # and #(x -9 )^2+(y +4 )^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?

1 Answer
Apr 6, 2016

The circles overlap but the small circle isn't contained in the big one. The biggest possible distance between 2 points in the 2 circles is #sqrt226+17~=32.033#

Explanation:

#(x+6)^2+(y-5)^2=64# => #r_1=8#, #C_1 (-6,5)#
#(x-9)^2+(y-4)^2=81# => #r_2=9#, #C_2 (9,4)#

#r_1+r_2=17#
#r_2-r_1=1#

#d_(C_1C_2)=sqrt((9+6)^2+(4-5)^2)=sqrt(225+1)=sqrt(226)~=15.033#

Since #r_2-r_1 < d_(C_1C_2) < r_1+r_2#
the circles overlap but circle 1 isn't contained in circle 2.

The greatest possible distance between 2 points in the 2 circles is
#=d_(C_1C_2)+r_1+r_2=sqrt226+8+9=sqrt226+17~=32.033#