Question

Circle A has a center at `(9 ,4 )` and a radius of `4`. Circle B has a center at `(-2 ,6 )` and a radius of `5`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap, smallest distance ≈ 2.18

Explanation:

What we have to do here is `color(blue)"compare"` the distance ( d ) between the centres of the circles 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)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 points"`

The 2 points here are (9 ,4) and (-2 ,6)

let `(x_1,y_1)=(9,4)" and " (x_2,y_2)=(-2,6)`

`d=sqrt((-2-9)^2+(6-4)^2)=sqrt(121+4)≈11.18`

Sum of radii = radius of A + radius of B = 4 + 5 = 9

Since sum of radii < d , then no overlap

smallest distance between them = d - sum of radii

`=11.18-9=2.18`
graph{(y^2-8y+x^2-18x+81)(y^2-12y+x^2+4x+15)=0 [-22.8, 22.8, -11.4, 11.41]}