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

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

Let's calculate the distance between the centers of Circle A and Circle B using the distance formula:

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

For Circle A with center ( (2, 4) ) and Circle B with center ( (9, 3) ), the distance is:

[ \text{Distance} = \sqrt{(9 - 2)^2 + (3 - 4)^2} ] [ \text{Distance} = \sqrt{(7)^2 + (-1)^2} ] [ \text{Distance} = \sqrt{49 + 1} ] [ \text{Distance} = \sqrt{50} ]

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

[ \text{Sum of radii} = \text{radius of Circle A} + \text{radius of Circle B} ] [ \text{Sum of radii} = 5 + 1 = 6 ]

Since ( \sqrt{50} > 6 ), the circles do not overlap.

To find the smallest distance between them, subtract the sum of the radii from the distance between their centers:

[ \text{Smallest distance} = \sqrt{50} - 6 ]

[ \text{Smallest distance} = \sqrt{50} - 6 \approx 0.35 ]

So, the smallest distance between the circles is approximately ( 0.35 ) units.

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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.

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- Circle A has a center at #(4 ,-1 )# and a radius of #3 #. Circle B has a center at #(-3 ,6 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(5 ,-2 )# and a radius of #2 #. Circle B has a center at #(4 ,6 )# and a radius of #3 #. Do the circles overlap? If not what is the smallest distance between them?
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