# Circle A has a center at (-5 ,8 ) and a radius of 4 . Circle B has a center at (-3 ,3 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Dec 15, 2016

$\text{circles overlap}$

#### Explanation:

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

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

$\text{the 2 points here are " (-5,8)" and } \left(- 3 , 3\right)$

$\text{let " (x_1,y_1)=(-5,8)" and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 3\right)$

d=sqrt((-3+5)^2+(3-8)^2)=sqrt(4+25)=sqrt29≈5.385

$\text{sum of radii = radius of A + radius of B = 4+4= 8}$

#Since sum of radii > d , then circles overlap.
graph{(y^2-16y+x^2+10x+73)(y^2-6y+x^2+6x+2)=0 [-28.48, 28.47, -14.24, 14.24]}