Circle A has a radius of 2  and a center of (3 ,1 ). Circle B has a radius of 4  and a center of (8 ,5 ). If circle B is translated by <-4 ,-1 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Jul 29, 2017

$\text{circles overlap}$

Explanation:

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

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

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

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

$\text{under a translation } \left(\begin{matrix}- 4 \\ - 1\end{matrix}\right)$

$\left(8 , 5\right) \to \left(8 - 4 , 5 - 1\right) \to \left(4 , 4\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

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

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)|)))

$\left({x}_{1} , {y}_{1}\right) = \left(3 , 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 4\right)$

$d = \sqrt{{\left(4 - 3\right)}^{2} + {\left(4 - 1\right)}^{2}} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162$

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

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