Question

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?

Answer 1

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 `((2),(-1))`

(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]}