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

Aug 6, 2018

$\text{circle B inside circle A}$

#### Explanation:

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

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

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

• " if difference of radii"> d" one circle inside other"

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

$\text{under the translation } < 2 , - 1 >$

$\left(1 , 4\right) \to \left(1 + 2 , 4 - 1\right) \to \left(3 , 3\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

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

$d = \sqrt{{\left(3 - 3\right)}^{2} + {\left(3 - 2\right)}^{2}} = \sqrt{1} = 1$

$\text{sum of radii } = 5 + 2 = 7$

$\text{difference of radii } = 5 - 2 = 3$

$\text{since difference of radii">d" circle B inside circle A}$
graph{((x-3)^2+(y-2)^2-25)((x-3)^2+(y-3)^2-4)=0 [-40, 40, -20, 20]}