Circle A has a center at #(3 ,7 )# and a radius of #2 #. Circle B has a center at #(4 ,-2 )# and a radius of #6 #. Do the circles overlap? If not, what is the smallest distance between them?

To determine if the circles overlap, we need to find 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 their radii, then the circles do not overlap. Otherwise, they overlap.

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Let's calculate the distance between the centers of Circle A $(3, 7)$ and Circle B $(4, -2)$:

$d = \sqrt{(4 - 3)^2 + (-2 - 7)^2}$ $d = \sqrt{1^2 + (-9)^2}$ $d = \sqrt{1 + 81}$ $d = \sqrt{82}$

The sum of the radii of Circle A and Circle B is $2 + 6 = 8$.

Since $\sqrt{82} > 8$, the circles do not overlap.

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

$\text{Smallest distance} = \sqrt{82} - 8$