Question

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?

Answer 1

circles overlap.

Explanation:

What we have to do here is `color(blue)"compare"` the distance (d) between the centres to the `color(blue)"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 `((2),(-1))`

`(1,7)to(1+2,7-1)to(3,6)`

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.

`rArrd=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]}