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

They do not overlap. The shortest distance is

Here is the graph of the circles:

The circles do not overlap.

The distance, d, between the centers is:

To find the shortest distance between them subtract the radii of both circles from d:

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To determine if the circles overlap, we need to compare the distance between their centers to the sum of their radii.

The distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by the distance formula: [d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]

For Circle A with center ((2, -1)) and Circle B with center ((-3, 6)), the distance between their centers is: [d = \sqrt{(-3 - 2)^2 + (6 - (-1))^2} = \sqrt{(-5)^2 + (7)^2} = \sqrt{25 + 49} = \sqrt{74}]

The sum of their radii is (3 + 1 = 4).

Since (\sqrt{74}) is greater than (4), the circles do not overlap.

To find the smallest distance between them, we subtract the sum of their radii from the distance between their centers: [ \text{Smallest distance} = \sqrt{74} - 4]

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