Circle A has a radius of 6  and a center of (2 ,5 ). Circle B has a radius of 3  and a center of (1 ,7 ). If circle B is translated by <3 ,1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 2, 2018

$\text{circles overlap}$

Explanation:

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

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

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

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

$\text{under the translation } < 3 , 1 >$

$\left(1 , 7\right) \to \left(1 + 3 , 7 + 1\right) \to \left(4 , 8\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(2,5)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 8\right)$

$d = \sqrt{{\left(4 - 2\right)}^{2} + {\left(8 - 5\right)}^{2}} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61$

$\text{sum of radii } = 6 + 3 = 9$

$\text{Since sum of radii">d" then circles overlap}$
graph{((x-2)^2+(y-5)^2-36)((x-4)^2+(y-8)^2-9)=0 [-40, 40, -20, 20]}