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Circle A has a center at #(9 ,-2 )# and a radius of #1 #. Circle B has a center at #(1 ,3 )# and a radius of #3 #. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

The circles do not overlap. There is a distance of (aprrox.) #5.434# between them.

The distance from the center of A at #(9,-2)# and the center of B at #(1,3)# is
#color(white)("XXX")d=sqrt((9-1)^2+(-2-3)^2)=sqrt(64+25)=sqrt(89)#

#color(white)("XXXx")~~9.434# (using a calculator)

Circle A with a radius of #1# covers #1# unit of this distance.
Circle B with a radius of #3# covers #3# units of this distance.

This leaves a minimum of #9.434-(1+3)=5.434# units uncovered between the two circles.

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Answer 2

Avery Allen

The circles do not overlap. The smallest distance between them is 1 unit.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.