Question

Circle A has a center at `(8 ,4 )` and a radius of `3`. Circle B has a center at `(-2 ,-2 )` and a radius of `4`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap , ≈ 4.662

Explanation:

What we have to do here is to 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)=(8,4)" and " (x_2,y_2)=(-2,-2)`

`d=sqrt((-2-8)^2+(-2-4)^2)=sqrt136≈11.662`

radius of A + radius of B = 3 + 4 = 7

Since sum of radii < d , then no overlap

and smallest distance between them = 11.662 - 7 = 4.662
graph{(y^2-8y+x^2-16x+71)(y^2+4y+x^2+4x-8)=0 [-20, 20, -10, 10]}