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

Jan 23, 2017

no overlap, smallest distance ≈ 1.47 units

#### Explanation:

What we have to do here is $\textcolor{b l u e}{\text{compare}}$ the distance (d) between the centres of the circles to the $\textcolor{b l u e}{\text{sum of radii}}$

• If sum of radii > d, then circles overlap

• If sum of radii < d, then no overlap

To calculate d, use the $\textcolor{b l u e}{\text{distance formula}}$

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

The 2 points here are (5 ,-2) and (1 ,-4)

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

d=sqrt((1-5)^2+(-4+2)^2)=sqrt(16+4)=sqrt20≈4.47

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

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

smallest distance between circles = d - sum of radii

$= 4.47 - 3 = 1.47$
graph{(y^2+4y+x^2-10x+25)(y^2+8y+x^2-2x+16)=0 [-10, 10, -5, 5]}