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?

Answer 1

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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Answer 2
To determine if circle B overlaps circle A after being translated by <1, 3>, we need to calculate the distance between the centers of the two circles after the translation and compare it to the sum of their radii. 1. Translate the center of circle B by <1, 3>: New center of circle B: (2 + 1, 4 + 3) = (3, 7) 2. Calculate the distance between the centers of circle A and the translated circle B: \( \text{Distance} = \sqrt{(6 - 3)^2 + (5 - 7)^2} = \sqrt{9 + 4} = \sqrt{13} \) 3. Check if the distance is less than the sum of the radii: \( \sqrt{13} < 2 + 3 = 5 \) Since \( \sqrt{13} < 5 \), the circles do overlap after the translation. If the circles did not overlap, the minimum distance between points on both circles could be found by subtracting the sum of their radii from the distance between their centers: \( \text{Minimum distance} = \sqrt{13} - 5 \).
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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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