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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The 2 points here are (4 ,-1) and (-3 ,2)

Sum of radii = radius of A + radius of B = 5 + 2 = 7

Since sum of radii < d , then no overlap

smallest distance between them = d - sum of radii

To determine whether 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 greater than the sum of the radii, 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 distance formula:

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

Given that the center of Circle A is $(4, -1)$ and the center of Circle B is $(-3, 2)$, and their radii are 5 and 2 respectively, we can calculate the distance between their centers.

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

$d = \sqrt{(-7)^2 + (3)^2}$

$d = \sqrt{49 + 9}$

$d = \sqrt{58}$

Now, we compare the distance between the centers to the sum of their radii:

$\text{Sum of radii} = 5 + 2 = 7$

Since $\sqrt{58} > 7$, 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:

$\text{Smallest distance} = \sqrt{58} - 7$

$\text{Smallest distance} = \sqrt{58} - 7$

$\text{Smallest distance} \approx 1.403$

So, the smallest distance between the circles is approximately $1.403$ units.