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

Feb 2, 2016

There is no overlap.

#### Explanation:

We can find the coordinates of the endpoints of the horizontal and vertical diameters of the circles by adding and subtracting the radius from the coordinates of the circles' centres.

Circle A has a horizontal diameter that runs from $\left(0 , - 2\right)$ to $\left(4 , - 2\right)$ and a vertical diameter that runs from $\left(2 , - 4\right)$ to $\left(2 , 0\right)$.

Circle A therefore has the domain $\left\{0 , 4\right\}$ and range $\left\{- 4 , 0\right\}$

Circle B has a horizontal diameter that runs from $\left(- 3 , - 4\right)$ to $\left(5 , - 4\right)$ and a vertical diameter that runs from $\left(1 , - 8\right)$ to $\left(1 , 0\right)$

Circle B has domain $\left\{- 3 , 5\right\}$ and range $\left\{- 8 , 0\right\}$

The domain of circle A is completely within the domain of circle B, and likewise the range of circle A is completely within the range of circle B, so circle A falls completely within circle B.