Circle A has a radius of #2 # and a center of #(6 ,5 )#. Circle B has a radius of #3 # and a center of #(2 ,4 )#. If circle B is translated by #<1 ,3 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

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Answer 1 · 2017-10-10T18:04:21

Circle B overlaps circle A after translation.

If circle B is translated by <#1,3#>, the center will be (#2+1,4+3#)=(#3,7#).
Let circle B' has a radius of #3# and a center of (#3,7#).

The distance #d# between the center of circle A and that of circle B' is:
#d=sqrt((3-6)^2+(7-5)^2)=sqrt(13)#

Let #r_a# and #r_b# to the radius of circle A and circle B(and B') respectively. #r_a=2, r_b=3#.

This satisfies the inequation:
#abs(r_a-r_b)< d< r_a+r_b#

Therefore circle A and circle B' (translated circle B) do neither circumscribe nor inscribe. They overlap.
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The figure is cited from http://examist.jp/mathematics/figure-circle/two-circle/ (Japanese)