Circle A has a center at #(2 ,2 )# and a radius of #5 #. Circle B has a center at #(12 ,8 )# and a radius of #1 #. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

The circles do not overlap.
There is a minimum distance of #2sqrt(34)-6~~5.66# units between them.

The length of a line segment joining the centers of the two circles is #color(white)("XXX")sqrt((12-2)^2+(8-2)^2) = sqrt(10^2+6^2) =2sqrt(34)#
The distance from the center of Circle A to the edge of Circle A along the line segment joining the two circles is #5# (the radius of A).
The distance from the edge of A to the center of Circle B along the line segment joining the centers is #2sqrt(34)-5#
The distance from the center of Circle B to the edge of Circle B along the line segment joining the centers is #1# (the radius of B).
The distance between the edges of the two circles is #color(white)("XXX")(2sqrt(34)-5)-1#
#color(white)("XXX")=2sqrt(34)-6#
#color(white)("XXX")~~5.66# (in whatever nits are being used)
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Answer 2

To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. The distance between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the formula:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

For Circle A with center ((2, 2)) and Circle B with center ((12, 8)):

[ \text{Distance} = \sqrt{(12 - 2)^2 + (8 - 2)^2} = \sqrt{100 + 36} = \sqrt{136} \approx 11.66 ]

The sum of their radii is (5 + 1 = 6). Since the distance between their centers ((\sqrt{136})) is greater than the sum of their radii, the circles do not overlap.

The smallest distance between them is the difference between the distance between their centers and the sum of their radii:

[ \text{Smallest Distance} = \text{Distance} - (\text{Radius of Circle A} + \text{Radius of Circle B}) ]

[ \text{Smallest Distance} = \sqrt{136} - (5 + 1) = \sqrt{136} - 6 \approx 5.66 ]

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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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