# Circle A has a center at (1 ,7 ) and a radius of 1 . Circle B has a center at (-3 ,-2 ) and a radius of 2 . Do the circles overlap? If not, what is the smallest distance between them?

Aug 10, 2017

$\text{no overlap } , \approx 6.849$

#### Explanation:

what we have to do here is $\textcolor{b l u e}{\text{compare }}$ the distance 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"

$\text{to calculate d use the "color(blue)"distance formula}$

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

$\left({x}_{1} , {y}_{1}\right) = \left(1 , 7\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 3 , - 2\right)$

$d = \sqrt{{\left(- 3 - 1\right)}^{2} + {\left(- 2 - 7\right)}^{2}} = \sqrt{16 + 81} \approx 9.849$

$\text{sum of radii } = 1 + 2 = 3$

$\text{since sum of radii"< d" then no overlap}$

$\text{smallest distance "=d-" sum of radii}$

$\textcolor{w h i t e}{s m a l \le s t \mathrm{di} s \tan c e} = 9.849 - 3 = 6.849$
graph{((x-1)^2+(y-7)^2-1)((x+3)^2+(y+2)^2-4)=0 [-20, 20, -10, 10]}