Circle A has a radius of 2  and a center at (3 ,6 ). Circle B has a radius of 4  and a center at (2 ,3 ). 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?

Jul 15, 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"

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

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

$B \left(2 , 3\right) \to \left(2 - 2 , 3 + 1\right) \to \left(0 , 4\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(3,6)" and } \left({x}_{2} , {y}_{2}\right) = \left(0 , 4\right)$

$d = \sqrt{{\left(0 - 3\right)}^{2} + {\left(4 - 6\right)}^{2}} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61$

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

$\text{Since sum of radii"> d" then circles overlap}$
graph{((x-3)^2+(y-6)^2-4)((x-0)^2+(y-4)^2-16)=0 [-20, 20, -10, 10]}