# Circle A has a radius of 3  and a center of (3 ,3 ). 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?

Jul 23, 2016

circles overlap.

#### Explanation:

What we have to do here is compare the distance (d ) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The first step is to calculate the 'new' coordinates of the centre of circle B under the given translation. Note the circle remains a circle but it's position changes.

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

(1 ,7) → (1+2 ,7-1) → B(3 ,6)

To calculate d , note that the centres A(3 ,3) and B(3 ,6) have the same x-coordinate and so d is just the difference in the y-coordinates.

Hence d = 6 - 3 = 3

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

Since sum of radii > d , then circles overlap.
graph{(y^2-6y+x^2-6x+9)(y^2-12y+x^2-6x+20)=0 [-40, 40, -20, 20]}