# Circle A has a center at (7 ,-2 ) and a radius of 2 . Circle B has a center at (4 ,2 ) and a radius of 4 . Do the circles overlap? If not, what is the smallest distance between them?

Jun 15, 2016

circles overlap

#### Explanation:

What we have to do here is to 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 $\left({x}_{1} , {y}_{1}\right) = \left(7 , - 2\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 2\right)$

$d = \sqrt{{\left(4 - 7\right)}^{2} + {\left(2 + 2\right)}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$

radius of A + radius of B = 2 + 4 = 6

Since sum of radii > d , then circles overlap
graph{(y^2+4y+x^2-14x+49)(y^2-4y+x^2-8x+4)=0 [-11.25, 11.25, -5.625, 5.625]}