Question

Circle A has a center at `(-2 ,-1 )` and a radius of `3`. Circle B has a center at `(-8 ,3 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

no overlap, ≈ 3.211

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 no overlap

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 (-2 ,-1) and (-8 ,3) the centres of the circles.

let `(x_1,y_1)=(-2,-1)" and " (x_2,y_2)=(-8,3)`

`d=sqrt((-8+2)^2+(3+1)^2)=sqrt(36+16)=sqrt52≈7.211`

sum of radii = radius of A + radius of B = 3 + 1 = 4

Since sum of radii < d , then there is no overlap.

smallest distance = d - sum of radii = 7.211 - 4 = 3.211
graph{(y^2+2y+x^2+4x-4)(y^2-6y+x^2+16x+72)=0 [-10, 10, -5, 5]}