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

Feb 14, 2017

no overlap,≈ 1.657

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

The 2 points here are (3 ,5) and (-1 ,1)

let $\left({x}_{1} , {y}_{1}\right) = \left(3 , 5\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 1 , 1\right)$

d=sqrt((-1-3)^2+(1-5)^2)=sqrt(16+16)=sqrt32≈5.657

sum of radii = radius of A + radius of B = 1 + 3 = 4

Since sum of radii < d, then no overlap of circles.

smallest distance between them = d - sum of radii

$= 5.657 - 4 = 1.657 \text{ units}$
graph{(y^2-10y+x^2-6x+33)(y^2-2y+x^2+2x-7)=0 [-20, 20, -10, 10]}