Question

Circle A has a radius of `1` and a center of `(1 ,2 )`. Circle B has a radius of `2` and a center of `(5 ,1 )`. If circle B is translated by `<-2 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circle B overlaps circle A

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

The first step,however, is to find the coordinates of the centre of circle B under the translation.

Under a translation of `((-2),(3))`

B(5 ,1) → (5-2 ,1+3) → B(3 ,4)

To calculate d, use the `color(blue)"distance formula"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

Here the 2 points are A(1 ,2) and B(3 ,4)

let `(x_1,y_1)=(1,2)" and " (x_2,y_2)=(3,4)`

`d=sqrt((3-1)^2+(4-2)^2)=sqrt(4+4)=sqrt8≈2.828`

Sum of radii = radius of A + radius of B = 1 + 2 = 3

Since sum of radii > d , then circles overlap.
graph{(y^2-4y+x^2-2x+4)(y^2-8y+x^2-6x+21)=0 [-13.86, 13.86, -6.92, 6.94]}