# Circle A has a radius of 4  and a center of (6 ,1 ). Circle B has a radius of 2  and a center of (5 ,3 ). If circle B is translated by <-2 ,2 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

May 31, 2018

$\text{circles overlap}$

#### Explanation:

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

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

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

$\text{before calculating d we require to find the new centre of}$
$\text{B under the given translation}$

$\text{under the translation } < - 2 , 2 >$

$\left(5 , 3\right) \to \left(5 - 2 , 3 + 2\right) \to \left(3 , 5\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(6,1)" and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 5\right)$

$d = \sqrt{{\left(3 - 6\right)}^{2} + {\left(5 - 1\right)}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$

$\text{sum of radii } = 4 + 2 = 6$

$\text{since sum of radii">d" then circles overlap}$
graph{((x-6)^2+(y-1)^2-16)((x-3)^2+(y-5)^2-4)=0 [-10, 10, -5, 5]}