# Circle A has a radius of 4  and a center of (8 ,5 ). Circle B has a radius of 2  and a center of (6 ,1 ). 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 26, 2017

$\text{circles overlap}$

#### Explanation:

$\text{what we have to do here is compare the distance (d )}$
$\text{between the centres of the circles to the}$
$\textcolor{b l u e}{\text{sum of the radii}}$

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

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

Before calculating d we require to find the coordinates of the new centre of B under the given translation which does not change the shape of the circle only its position.

$\text{under a translation } \left(\begin{matrix}3 \\ 1\end{matrix}\right)$

$\left(6 , 1\right) \to \left(6 + 3 , 1 + 1\right) \to \left(9 , 2\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

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

color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)color(white)(2/2)|)))

$\left({x}_{1} , {y}_{1}\right) = \left(8 , 5\right) , \left({x}_{2} , {y}_{2}\right) = \left(9 , 2\right)$

$d = \sqrt{{\left(9 - 8\right)}^{2} + {\left(2 - 5\right)}^{2}} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162$

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

$\text{since sum of radii > d then circles overlap}$
graph{(y^2-10y+x^2-16x+73)(y^2-4y+x^2-18x+81)=0 [-20, 20, -10, 10]}