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

Sep 13, 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 it's position.

$\text{under a translation } \left(\begin{matrix}2 \\ - 1\end{matrix}\right)$

$\left(1 , 4\right) \to \left(3 , 3\right) \leftarrow \textcolor{red}{\text{ new centre of B}}$

$\text{since "(5,3)" and "(3,3)" have the same y-coordinate then}$
$\text{d is the difference of the x-coordinates}$

$\Rightarrow d = 5 - 3 = 2$

$\text{sum of radii } = 4 + 3 = 7$

$\text{since sum of radii ">d" then circles overlap}$
graph{((x-5)^2+(y-3)^2-16)((x-3)^2+(y-3)^2-9)=0 [-20, 20, -10, 10]}