# Circle A has a radius of 6  and a center of (8 ,5 ). Circle B has a radius of 3  and a center of (6 ,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?

May 12, 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 $\textcolor{b l u e}{\text{sum of their radii}}$

• "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 give translation, which does not change the shape of the circle only it's position.

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

$\left(6 , 7\right) \to \left(6 + 3 , 7 + 1\right) \to \left(9 , 8\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where $\left({x}_{1} , {y}_{1}\right) , \left({x}_{2} , {y}_{2}\right) \text{ are 2 coordinate points}$

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

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

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

$\text{Since sum of radii" > d" then circles overlap}$
graph{(y^2-10y+x^2-16x+53)(y^2-16y+x^2-18x+136)=0 [-22.8, 22.81, -11.4, 11.4]}