Circle A has a radius of #3 # and a center of #(5 ,4 )#. Circle B has a radius of #4 # and a center of #(7 ,2 )#. If circle B is translated by #<3 ,2 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2017-07-06T14:08:38

The circles overlap

Let the center of the circle #A# be #O_A=(5,4)#

and the center of the circle #B# be #O_B=(7,2)#

The center after translation is

#O_B'=(7,2)+ <3,2> =(10,4)#

The distance between the centers is

#O_AO_B'=sqrt((10-5)^2+(4-4)^2)#

#=5#

The sum of the radii is

#r_A+r_B=3+4=7#

As ,

#O_AO_B'<(r_A+r_B)#

The circles overlap

graph{((x-5)^2+(y-4)^2-9)((x-7)^2+(y-2)^2-16)((x-10)^2+(y-4)^2-16)=0 [-6, 16.5, -2.185, 9.065]}