Circle A has a radius of #2 # and a center at #(3 ,1 )#. Circle B has a radius of #4 # and a center at #(8 ,3 )#. If circle B is translated by #<-4 ,-1 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

1 Answer
turksvids
Jan 13, 2018

They don't overlap. The minimum distance is #2-sqrt(2)#.

Explanation:

The new center of circle B is #(4,2)#.

The distance between the centers of the two circles is

#sqrt((4-3)^2 + (2-1)^2)= sqrt(2)#

Circle B is the larger circle and all points on the circle are 4 units from its center.

The radius of circle A is 2 and the center of A is #sqrt(2)# units from the center of Circle B so the farthest any point on A can be from the center of B is #2+sqrt(2)#. Since #2+sqrt(2)<4# no point on A overlaps any point on B.

The minimum distance between points on A and B is actually the radius of B, which is 4, minus the farthest any point on A can be from the center of B, which is #2+sqrt(2)#:

#4-(2+sqrt(2)) = 2-sqrt(2)#

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