# Circle A has a radius of 3  and a center of (2 ,4 ). Circle B has a radius of 2  and a center of (4 ,7 ). If circle B is translated by <2 ,-3 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Oct 1, 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 $\textcolor{b l u e}{\text{sum of the radii}}$

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

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

$\text{before calculating d we require to find the 'new' }$
$\text{centre of B under the given translation which does}$
$\text{not change the shape of the circle only it's position}$

$\text{under a translation } \left(\begin{matrix}2 \\ - 3\end{matrix}\right)$

$\left(4 , 7\right) \to \left(4 + 2 , 7 - 3\right) \to \left(6 , 4\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

$\text{since the centres have the same y-coordinate then}$
$\text{d is the difference in the x-coordinates}$

$\Rightarrow d = 6 - 2 = 4$

$\text{sum of radii } = 3 + 2 = 5$

$\text{since sum of radii">d" then circles overlap}$
graph{((x-2)^2+(y-4)^2-9)((x-6)^2+(y-4)^2-4)=0 [-10, 10, -5, 5]}