Circle A has a radius of 3  and a center of (3 ,2 ). Circle B has a radius of 5  and a center of (1 ,7 ). 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?

Jan 12, 2017

circles overlap.

Explanation:

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

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

$\left(1 , 7\right) \to \left(1 + 2 , 7 - 1\right) \to \left(3 , 6\right)$

Note that the centres (3 ,2) and (3 ,6) have the same x-coordinate and so d is just the difference in the y-coordinates.

$\Rightarrow d = 6 - 2 = 4$

sum of radii = radius of A + radius of B = 3 + 5 = 8

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-6x+4)(y^2-12y+x^2-6x+20)=0 [-40, 40, -20, 20]}