Question

Circle A has a radius of `2` and a center of `(8 ,2 )`. Circle B has a radius of `4` and a center of `(5 ,3 )`. If circle B is translated by `<-1 ,5 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap, min distance ≈ 1.211 units

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before doing this we require to find the new centre of B under the translation, which does not change the shape of the circle only it's position.

Under a translation `((-1),(5))`

(5 ,3) → (5-1,3+5) → (4 ,8) is the new centre of B.

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"`

The 2 points here are (8 ,2) and (4 ,8) the centres of the circles.

let `(x_1,y_1)=(8,2)" and " (x_2,y_2)=(4,8)`

`d=sqrt((4-8)^2+(8-2)^2)=sqrt(16+36)=sqrt52≈7.211`

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

Since sum of radii < d , then no overlap

minimum distance between points = d - sum of radii

=7.211 - 6 = 1.211 units
graph{(y^2-4y+x^2-16x+64)(y^2-16y+x^2-8x+64)=0 [-40, 40, -20, 20]}