# Circle A has a center at (2 ,5 ) and an area of 16 pi. Circle B has a center at (8 ,1 ) and an area of 3 pi. Do the circles overlap? If not, what is the shortest distance between them?

May 22, 2018

$\text{no overlap } \approx 1.479$

#### Explanation:

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

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

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

$\text{to find the radii use area (A) of circle formula}$

•color(white)(x)A=pir^2

$\text{circle A } \to \pi {r}^{2} = 16 \pi \Rightarrow r = 4$

$\text{circle B } \to \pi {r}^{2} = 3 \pi \Rightarrow r = \sqrt{3}$

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

•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

$\text{let "(x_1,y_1)=(2,5)" and } \left({x}_{2} , {y}_{2}\right) = \left(8 , 1\right)$

$d = \sqrt{{\left(8 - 2\right)}^{2} + {\left(1 - 5\right)}^{2}} = \sqrt{36 + 16} = \sqrt{52} \approx 7.211$

$\text{sum of radii } = 4 + \sqrt{3} \approx 5.732$

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

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

$\textcolor{w h i t e}{\times \times \times \times \times \times \times} = 7.211 - 5.732 = 1.479$
graph{((x-2)^2+(y-5)^2-16)((x-8)^2+(y-1)^2-3)=0 [-10, 10, -5, 5]}