# Circle A has a center at (-1 ,-4 ) and a radius of 3 . 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 22, 2016

no overlap , ≈ 1.325

#### Explanation:

What we have to do here is 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 $\textcolor{b l u e}{\text{distance formula}}$

color(red)(bar(ul(|color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|))
where $\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are 2 coordinate points.}$

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

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

d=sqrt((-3-(-1))^2+(2-(-4))^2)=sqrt(4+36)≈6.325

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

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

smallest distance between them = d - sum of radii

$= 6.325 - 5 = 1.325$
graph{(y^2-4y+x^2+6x+9)(y^2+8y+x^2+2x+8)=0 [-16.22, 16.26, -8.15, 8.08]}