Question

Circle A has a center at `(-1 ,2 )` and a radius of `2`. Circle B has a center at `(5 ,-4 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

no overlap , ≈ 5.485

Explanation:

What we have to do here is compare the distance (d ) between the centres with the 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 points"`

let `(x_1,y_1)=(-1,2)" and " (x_2,y_2)=(5,-4)`

`d=sqrt((5+1)^2+(-4-2)^2)=sqrt72≈8.485`

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

Since sum of radii < d , then no overlap

smallest distance = 8.485 - 3 = 5.485
graph{(y^2-4y+x^2+2x+1)(y^2+8y+x^2-10x+40)=0 [-10, 10, -5, 5]}