# Circle A has a center at (-2 ,-7 ) and a radius of 2 . Circle B has a center at (-3 ,2 ) and a radius of 5 . Do the circles overlap? If not, what is the smallest distance between them?

Nov 25, 2016

$\textcolor{b l u e}{\text{Thus smallest distance between them is exactly } \sqrt{82} - 7}$

#### Explanation:

Let point 1 $\to {P}_{1} = \left({x}_{1} , {y}_{1}\right) = \left(- 2 , - 7\right)$

Let point 2 $\to {P}_{2} = \left({x}_{2} , {y}_{2}\right) = \left(- 3 , 2\right)$

Let distance between centres be $c$

$\textcolor{b l u e}{\text{Sum of radii = 2 + 5 = 7}}$

Distance between centres $\to c = {P}_{2} - {P}_{1}$

So by Pythagoras:

$\implies c = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$c = \sqrt{{\left[\textcolor{w h i t e}{.} \left(- 3\right) - \left(- 2\right) \textcolor{w h i t e}{.}\right]}^{2} + {\left[\textcolor{w h i t e}{.} 2 - \left(- 7\right) \textcolor{w h i t e}{.}\right]}^{2}}$

$c = \sqrt{1 + 81} = 9.055$ to 3 decimal places

$\textcolor{b l u e}{\text{Distance centre to centre = 9.055 to 3 decimal places}}$

$\textcolor{b r o w n}{\text{Smallest distance between is (centre to centre) - (sum of radii)}}$

$\textcolor{b l u e}{\text{Thus smallest distance between them is exactly } \sqrt{82} - 7}$