Circle A has a center at #(1 ,-4 )# and a radius of #2 #. Circle B has a center at #(9 ,3 )# and a radius of #5 #. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

The circles do not overlap;
they are separated by a minimum distance of #1.6# units

The distance between the centers of the circles: #color(white)("XXX")d=sqrt((1-9)^2+(-4-3)^2) =sqrt(113)~~10.6#

Circle A covers 2 (units) of the distance along the line segment from the center of A to the center of B.

Circle B covers 5 (units) of the distance along the line segment form the center of B to the center of A.

A distance of #color(white)("XXX")10.6 -(2+7)=1.6# along the line segment joining the two centers is not covered by either circle.
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Answer 2

The circles do not overlap. The distance between their centers can be calculated using the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Substituting the given coordinates: Distance = √((9 - 1)^2 + (3 - (-4))^2) = √((8)^2 + (7)^2) = √(64 + 49) = √113 ≈ 10.63

The sum of the radii of the two circles is 2 + 5 = 7. Since the distance between the centers of the circles (10.63) is greater than the sum of their radii (7), the circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii:

Smallest distance = Distance between centers - Sum of radii = 10.63 - 7 = 3.63

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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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