# Circle A has a radius of 3  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 <2 ,7 >, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Oct 2, 2016

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

Before calculating d, require to find the 'new' centre of circle B under the given transformation which does not change the shape of the circle only it's position.

Under a translation of $\left(\begin{matrix}2 \\ 7\end{matrix}\right)$

$\left(6 , 1\right) \to \left(6 + 2 , 1 + 7\right) \to \left(8 , 8\right) \leftarrow \text{ new centre of B }$

To calculate d, in this case since the coordinates of the centres (8 ,5) and (8 ,8) have the same x-coordinate- that is they lie on the same vertical line, then d is the difference in the y-coordinates.

$\Rightarrow d = 8 - 5 = 3$

Sum of radii = radius of A + radius of B = 3 + 2 + 5

Since sum of radii > d , then the circles overlap
graph{(y^2-10y+x^2-16x+80)(y^2-16y+x^2-16x+124)=0 [-28.48, 28.48, -14.22, 14.26]}