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

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The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centersThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance betweenThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

SmThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distanceThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: DistanceThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance =The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrtThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrtThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((xThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 -The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 -The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - xThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 +The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 +The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (yThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 -The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 -The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - yThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2)The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) -The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance =The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3)The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 -The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) =The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrtThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 +The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9)The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) -The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 -The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 =The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrtThe circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance =The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58)The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrtThe circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 +The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2.The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (3The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2.58The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (3)^The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2.58.The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (3)^2The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2.58.The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (3)^2) The circles do not overlap. The smallest distance between them is the difference between the sum of their radii and the distance between their centers:

Smallest distance = sqrt((4 - (-3))^2 + (-1 - 2)^2) - (1 + 3) = sqrt(49 + 9) - 4 = sqrt(58) - 4 ≈ 2.58.The circles do not overlap. The smallest distance between them is the distance between their centers minus the sum of their radii.

Distance between centers: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Distance = sqrt((-3 - 4)^2 + (2 - (-1))^2) Distance = sqrt((-7)^2 + (3)^2) Distance = sqrt(49 + 9) Distance = sqrt(58)

Sum of radii: Sum = radius of circle A + radius of circle B Sum = 1 + 3 Sum = 4

Smallest distance between the circles: Smallest distance = Distance - Sum Smallest distance = sqrt(58) - 4 ≈ 4.16 - 4 = 0.16

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To determine if the circles overlap, we can calculate the distance between their centers and compare it to the sum of their radii. If the distance between the centers is less than the sum of their radii, then the circles overlap. Otherwise, they do not overlap, and the smallest distance between them is the difference between the distance between their centers and the sum of their radii.

Let's denote the centers of the circles A and B as ((x_1, y_1)) and ((x_2, y_2)) respectively, and their radii as (r_1) and (r_2) respectively.

For circle A:

Center coordinates: ((4, -1))

Radius: (r_1 = 1)

For circle B:

Center coordinates: ((-3, 2))

Radius: (r_2 = 3)

Using the distance formula, the distance between the centers is calculated as:

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

Substituting the given values:

[ \text{Distance} = \sqrt{(-3 - 4)^2 + (2 - (-1))^2} ] [ \text{Distance} = \sqrt{(-7)^2 + (3)^2} ] [ \text{Distance} = \sqrt{49 + 9} ] [ \text{Distance} = \sqrt{58} ]

Now, we compare the distance to the sum of the radii:

[ \text{Sum of radii} = r_1 + r_2 = 1 + 3 = 4 ]

Since ( \sqrt{58} ) is greater than 4, 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{Sum of radii} = \sqrt{58} - 4 ]

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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.

- A triangle has corners at #(1 ,4 )#, #(7 ,5 )#, and #(3 ,2 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(5 ,-2 )# and a radius of #2 #. Circle B has a center at #(1 ,-4 )# and a radius of #1 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(-4 ,-1 )# and a radius of #3 #. Circle B has a center at #(1 ,3 )# and a radius of #2 #. Do the circles overlap? If not what is the smallest distance between them?
- A line passes through #(5 ,0 )# and #(7 ,3 )#. A second line passes through #(3 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?
- A line passes through #(1 ,5 )# and #(8 ,7 )#. A second line passes through #(3 ,6 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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