Circle A has a radius of 2  and a center of (6 ,5 ). Circle B has a radius of 1  and a center of (3 ,4 ). If circle B is translated by <1 ,3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 22, 2018

$\text{circles overlap}$

Explanation:

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

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

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

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

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

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

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

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

$\text{let "(x_1,y_1)=(6,5)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 7\right)$

$d = \sqrt{{\left(4 - 6\right)}^{2} + {\left(7 - 5\right)}^{2}} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$

$\text{sum of radii } = 2 + 1 = 3$

$\text{Since sum of radii">d" then circles overlap}$
graph{((x-6)^2+(y-5)^2-4)((x-4)^2+(y-7)^2-1)=0 [-40, 40, -20, 20]}