# Circle A has a center at (2 ,1 ) and a radius of 4 . Circle B has a center at (4 ,7 ) and a radius of 3 . Do the circles overlap? If not what is the smallest distance between them?

Feb 4, 2016

The circles overlap because both the domains and the ranges overlap.

#### Explanation:

Work out the domain and range of each of the circles. If those overlap then the circles overlap.

The $x$ values of circle $A$ run from $2 - 4$ to $2 + 4$ so the domain is $\left\{- 2 , 6\right\}$

The $x$ values of circle $B$ run from $4 - 3$ to $4 + 3$ so the domain is $\left\{1 , 7\right\}$

The $y$ values of circle $A$ run from $1 - 4$ to $1 + 4$ so the range is $\left\{- 3 , 5\right\}$

The $y$ values for circle $B$ run from $7 - 3$ to $7 + 3$ so the range is $\left\{4 , 10\right\}$

The circles overlap because both the domains and the ranges overlap.