Circle A has a radius of #3 # and a center of #(2 ,7 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2017-06-28T14:41:44

The circles do not overlap and the minimum distance is #=1.08#

Circle #A#, center #O_A=(2,7)#

The equation of circle #A# is

#(x-2)^2+(y-7)^2=9#

Circle #B#, center #O_B=(6,1)#

The equation of circle #B# is

#(x-6)^2+(y-1)^2=4#

The center of circle #B'# after translation is

#(6,1)+(2,7)=(8,8)#

Circle #B'#, center #O_B'=(8,8)#

The equation of the circle after translation is

#(x-8)^2+(y-8)^2=4#

The distance #O_AO_B'# is

#=sqrt((8-2)^2+(8-7)^2)#

#=sqrt(36+1)#

#=sqrt37#

#=6.08#

This distance is greater than the sum of the radii

#O_AO_B'>r_A+r_B'#

So, the circles do not overlap and the minimum distance is

#=6.08-(2+3)#

#=1.08#
graph{((x-2)^2+(y-7)^2-9)((x-6)^2+(y-1)^2-4)((x-8)^2+(y-8)^2-4)=0 [-7.28, 18.03, -1.57, 11.09]}