# Circle A has a center at (2 ,12 ) and an area of 81 pi. Circle B has a center at (1 ,3 ) and an area of 36 pi. Do the circles overlap? If not, what is the shortest distance between them?

Sep 28, 2017

$\text{circles overlap}$

#### Explanation:

$\text{what we have to do here is to "color(blue)"compare}$
$\text{the distance (d) between the centres of the circles}$
$\text{with the "color(blue)"sum of the radii}$

• " if sum of radii ">d" then circles overlap"

• " if sum of radii "< d" then no overlap"

$\text{to calculate the radii of the circles use}$

• " area of circle "=pir^2larr" r is the radius"

$\textcolor{b l u e}{\text{circle A}} \textcolor{w h i t e}{x} \pi {r}^{2} = 81 \pi \Rightarrow r = 9$

$\textcolor{b l u e}{\text{circle B }} \pi {r}^{2} = 36 \pi \Rightarrow r = 6$

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

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

$\text{here "(x_1,y_1)=(1,3)" and } \left({x}_{2} , {y}_{2}\right) = \left(2 , 12\right)$

$d = \sqrt{{\left(2 - 1\right)}^{2} + {\left(12 - 3\right)}^{2}} = \sqrt{1 + 81} = \sqrt{82} \approx 9.055$

$\text{sum of radii } = 9 + 6 = 15$

$\text{since sum of radii ">d" then circles overlap}$
graph{((x-2)^2+(y-12)^2-81)((x-1)^2+(y-3)^2-36)=0 [-40, 40, -20, 20]}