# Circle A has a center at (-1 ,2 ) and a radius of 2 . Circle B has a center at (5 ,-4 ) and a radius of 1 . Do the circles overlap? If not what is the smallest distance between them?

Jun 13, 2016

no overlap , ≈ 5.485

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

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

let $\left({x}_{1} , {y}_{1}\right) = \left(- 1 , 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(5 , - 4\right)$

d=sqrt((5+1)^2+(-4-2)^2)=sqrt72≈8.485

radius of A + radius of B = 2 + 1 = 3

Since sum of radii < d , then no overlap

smallest distance = 8.485 - 3 = 5.485
graph{(y^2-4y+x^2+2x+1)(y^2+8y+x^2-10x+40)=0 [-10, 10, -5, 5]}