Question

Circle A has a radius of `4` and a center of `(5 ,3 )`. Circle B has a radius of `2` and a center of `(1 ,2 )`. If circle B is translated by `<2 ,4 >`, 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 of the circles 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 coordinates of the 'new' centre of circle B under the given translation. Under a translation the circle remains a circle but it's position changes.

Under a translation `((2),(4))`

(1 ,2) → (1+2 ,2+4) → new centre of B is (3 ,6)

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 (5 ,3) and (3 ,6) the centres of the circles.

let `(x_1,y_1)=(5,3)" and " (x_2,y_2)=(3,6)`

`d=sqrt((3-5)^2+(6-3)^2)=sqrt(4+9)=sqrt13≈3.606`

sum of radii = radius of A + radius of B = 4 + 2 = 6

Since sum of radii > d , then circles overlap
graph{(y^2-6y+x^2-10x+18)(y^2-12y+x^2-6x+41)=0 [-20, 20, -10, 10]}