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

The circles do not overlap.

The minimum distance between them is

The distance between the centers of the two circles is

Consider a line segment joining the two centers.

The distance from A's center to A's circumference is

The distance from B's center to B's circumference is

The sum of the radial distances is

by

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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. The formula to find the distance between two points (x1, y1) and (x2, y2) is:

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

For Circle A with center (3, 5) and Circle B with center (-1, 1):

Distance = √((-1 - 3)^2 + (1 - 5)^2) Distance = √((-4)^2 + (-4)^2) Distance = √(16 + 16) Distance = √32 Distance ≈ 5.657

Now, let's compare this distance to the sum of their radii:

Radius of Circle A = 1 Radius of Circle B = 4

Sum of Radii = 1 + 4 = 5

Since the distance between the centers (approximately 5.657) is greater than the sum of their radii (5), the circles overlap.

If the circles did not overlap, the smallest distance between them would be the difference between the distance between their centers and the sum of their radii. In this case, it would be approximately 0.657 units.

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