# 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?

Sep 27, 2016

no overlap, ≈ 0.616

#### 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 $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

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

let $\left({x}_{1} , {y}_{1}\right) = \left(4 , - 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 2\right)$

d=sqrt((-3-4)^2+(2+1)^2)=sqrt(49+9)=sqrt58≈7.616

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

$= 7.616 - 7 = 0.616$
graph{(y^2+2y+x^2-8x-8)(y^2-4y+x^2+6x+9)=0 [-20, 20, -10, 10]}