Two circles have the following equations #(x +3 )^2+(y -6 )^2= 64 # and #(x +7 )^2+(y +2 )^2= 9 #. 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 · 2017-09-29T17:10:53

The circles overlap

Let the circles be #C_1# and #C_2#

Let the radius of the circles be #r_1# and #r_2#

Let the distance between the centers be #=d#

The circles overlap, i.e,

#C_1nnC_2!=O/#

#hArr#

#d<(r_1+r_2)#

Here, we have

#d=sqrt((-7+3)^2+(-2+6)^2)=sqrt(16+16)=sqrt32#

#r_1=8#

#r_2=3#

#r_1+r_2=8+3=11#

#(r_1 +r_2) > d #

The circles overlap

graph{((x+3)^2+(y-6)^2-64)((x+7)^2+(y+2)^2-9)=0 [-32.46, 32.47, -16.23, 16.25]}