# Circle A has a center at (3 ,-1 ) and a radius of 6 . Circle B has a center at (-2 ,-4 ) and a radius of 3 . Do the circles overlap? If not, what is the smallest distance between them?

Sep 22, 2016

circles overlap.

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

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

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

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

d=sqrt((-2-3)^2+(-4+1)^2)=sqrt(25+9)≈5.831

sum of radii = radius of A + radius of B = 6 + 3 = 9

Since sum of radii > d , then the circles overlap
graph{(y^2+2y+x^2-6x-26)(y^2+8y+x^2+4x+11)=0 [-20, 20, -10, 10]}