Two circles have the following equations #(x +5 )^2+(y -2 )^2= 36 # and #(x +2 )^2+(y -1 )^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 · 2016-12-26T10:59:01

The circles overlap
The greatest possible distance is #=18.16#

We need the equation of a circle, center #(a,b)# and radius #r# is

#(x-a)^2+(y-b)^2=r^2#

The distance between 2 points, #(x_1,y_1)# and #(x_2,y_2)# is

#=sqrt((x_2-x_1)^2+(x_2-y_2)^2)#

We need to find the distance between the centres of the circles and compare this to the sum of the radii.

The centers are #C_A=(-5,2)# and #C_B=(-2,1)#

The distance between the centers is

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

#=sqrt(9+1)=sqrt10=3.16#

The sum of the radii is #=6+9=15#

Therefore,

#d<#sum of radii

so,

The circles overlap

The greatest possible distance is #=9+6+3.16=18.16#

graph{((x+5)^2+(y-2)^2-36)((x+2)^2+(y-1)^2-81)(y-2+1/3(x+5)) = 0 [-19.22, 9.27, -5.39, 8.86]}