Two circles have the following equations: #(x +6 )^2+(y -1 )^2= 49 # 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?

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Answer 1 · 2017-02-18T16:45:20

The circles overlap and the greatest distance is #=31.3#

We compare the sum of the radii to the distance between the centers,

Radius of first circle #r_1=7#

Radius of second circle #r_2=9#

#r_1+r_2=9+7=16#

The center of the first circle is #O=(-6,1)#

The center of the second circle is #O'=(9,4)#

The distance between the centers is

#d_(OO') = sqrt((9--6)^2+(4-1)^2)#

#=sqrt(225+9)#

#=sqrt234=15.3#

Therefore,

#d_(OO') < r_1+r_2#

So,

the circles overlap

The greatest distance is #=15.3+7+9=31.3#

graph{((x+2)^2+(y-1)^2-49)((x-9)^2+(y-4)^2-81)(y-4-1/5(x-9))=0 [-22.8, 22.83, -11.4, 11.4]}