Circle A has a center at (-2 ,-1 ) and a radius of 3 . Circle B has a center at (-8 ,3 ) and a radius of 1 . Do the circles overlap? If not what is the smallest distance between them?

Sep 13, 2016

no overlap, ≈ 3.211

Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance ( d) between the centres 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 $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points}$

here the 2 points are (-2 ,-1) and (-8 ,3) the centres of the circles.

let $\left({x}_{1} , {y}_{1}\right) = \left(- 2 , - 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 8 , 3\right)$

d=sqrt((-8+2)^2+(3+1)^2)=sqrt(36+16)=sqrt52≈7.211

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

Since sum of radii < d , then there is no overlap.

smallest distance = d - sum of radii = 7.211 - 4 = 3.211
graph{(y^2+2y+x^2+4x-4)(y^2-6y+x^2+16x+72)=0 [-10, 10, -5, 5]}