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

Answer 1

Aurora Ayala

The circles do not overlap. The minimum distance is #=1.3#

The sum of the radii of the circles is

#r_A+r_B=1+4=5#

The center of circle #B# after translation is

#=(5,3)+<-2,5>=(3,8)#

The distance between the centers is

#d=sqrt((3-1)^2+(8-2)^2)=sqrt(4+36)=sqrt40=6.3#

#d>r_A+r_B#

The circles do not overlap.

The minimum distance is

#=d-(r_A+r_B)=6.3-5=1.3#

graph{((x-1)^2+(y-2)^2-1)((x-3)^2+(y-8)^2-16)=0 [-12.58, 15.9, -3, 11.24]}

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Answer 2

Wyatt Anderson

No, Circle B does not overlap Circle A after being translated by <-2, 5>. The minimum distance between points on both circles is 2 units.

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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.

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

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