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

May 8, 2016

no overlap

#### Explanation:

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

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

$d = \sqrt{{\left(1 - \left(- 4\right)\right)}^{2} + {\left(- 1 - 2\right)}^{2}} = \sqrt{25 + 9}$

=sqrt34 ≈ 5.83

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

Since sum of radii < d , then no overlap
graph{(y^2-4y+x^2+8x+11)(y^2+2y+x^2-2x+1)=0 [-10, 10, -5, 5]}