Circle A has a radius of 3  and a center at (1 ,2 ). Circle B has a radius of 5  and a center at (3 ,7 ). 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?

Jun 2, 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"

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

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

$\left(3 , 7\right) \to \left(3 + 2 , 7 + 1\right) \to \left(5 , 8\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

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

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

$\text{sum of radii } = 3 + 5 = 8$

$\text{since sum of radii"> d" then circles overlap}$
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-8)^2-25)=0 [-20, 20, -10, 10]}