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

Mar 6, 2016

#### Explanation:

Mar 6, 2016

No overlap , distance ≈ 3.485

#### Explanation:

The first step is to calculate the distance between the centres using the $\textcolor{b l u e}{\text{ distance formula }}$

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

let$\left({x}_{1} , {y}_{1}\right) = \left(- 1 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(5 , - 4\right)$

$\Rightarrow d = \sqrt{{\left(5 + 1\right)}^{2} + {\left(- 4 - 2\right)}^{2}} = \sqrt{36 + 36}$

= sqrt72 = 6sqrt2≈ 8.485

now radius of A + radius of B = 2+3 = 5

since 5 < 8.485 there is no overlap.

and distance between them is 8.485 - 5 = 3.485