Circle A has a radius of #3 # and a center of #(2 ,7 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

The circles do not overlap and the minimum distance is #=1.08#

Circle #A#, center #O_A=(2,7)#
The equation of circle #A# is
#(x-2)^2+(y-7)^2=9#
Circle #B#, center #O_B=(6,1)#
The equation of circle #B# is
#(x-6)^2+(y-1)^2=4#
The center of circle #B'# after translation is
#(6,1)+(2,7)=(8,8)#
Circle #B'#, center #O_B'=(8,8)#

The equation of the circle after translation is

#(x-8)^2+(y-8)^2=4#
The distance #O_AO_B'# is
#=sqrt((8-2)^2+(8-7)^2)#
#=sqrt(36+1)#
#=sqrt37#
#=6.08#

This distance is greater than the sum of the radii

#O_AO_B'>r_A+r_B'#

So, the circles do not overlap and the minimum distance is

#=6.08-(2+3)#
#=1.08# graph{((x-2)^2+(y-7)^2-9)((x-6)^2+(y-1)^2-4)((x-8)^2+(y-8)^2-4)=0 [-7.28, 18.03, -1.57, 11.09]}
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Answer 2
No, Circle B does not overlap with Circle A after the translation by <2, 7>. The minimum distance between points on both circles can be found by calculating the distance between their centers and subtracting the sum of their radii. The distance between the centers of Circle A (2, 7) and Circle B (6, 1) after the translation is: \[ \sqrt{(6 - 2)^2 + (1 + 7)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5} \] The sum of their radii is \( 3 + 2 = 5 \). Therefore, the minimum distance between points on both circles is: \[ 4\sqrt{5} - 5 \]
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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.

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.

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.

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.

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