Circle A has a radius of 2  and a center at (3 ,1 ). Circle B has a radius of 4  and a center at (8 ,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?

May 26, 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"

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

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

$\left(8 , 3\right) \to \left(8 - 2 , 3 + 1\right) \to \left(6 , 4\right) \leftarrow \textcolor{red}{\text{new centre of B}}$

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

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

$\text{let "(x_1,y_1)=(6,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(3 , 1\right)$

$d = \sqrt{{\left(3 - 6\right)}^{2} + {\left(1 - 4\right)}^{2}} = \sqrt{9 + 9} = \sqrt{18} \approx 4.24$

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

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