# Circle A has a center at (6 ,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?

Dec 1, 2016

no overlap , ≈ 1.083

#### 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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

The 2 points here are (6 ,2) and (5 ,-4)

let $\left({x}_{1} , {y}_{1}\right) = \left(6 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(5 , - 4\right)$

d=sqrt((5-6)^2+(-4-2)^2)=sqrt(1+36)=sqrt37≈6.083

sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii < d , then no overlap

smallest distance between them = d - sum of radii

$= 6.083 - 5 = 1.083$
graph{(y^2-4y+x^2-12x+36)(y^2+8y+x^2-10x+32)=0 [-14.24, 14.24, -7.12, 7.12]}