# Circle A has a center at (9 ,4 ) and a radius of 4 . Circle B has a center at (-2 ,6 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Nov 26, 2016

no overlap, smallest distance ≈ 2.18

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance ( d ) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of the radii.}}$

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

To calculate d, use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 points}$

The 2 points here are (9 ,4) and (-2 ,6)

let $\left({x}_{1} , {y}_{1}\right) = \left(9 , 4\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 2 , 6\right)$

d=sqrt((-2-9)^2+(6-4)^2)=sqrt(121+4)≈11.18

Sum of radii = radius of A + radius of B = 4 + 5 = 9

Since sum of radii < d , then no overlap

smallest distance between them = d - sum of radii

$= 11.18 - 9 = 2.18$
graph{(y^2-8y+x^2-18x+81)(y^2-12y+x^2+4x+15)=0 [-22.8, 22.8, -11.4, 11.41]}