Question

Circle A has a center at `(6 ,2 )` and a radius of `2`. Circle B has a center at `(5 ,-4 )` and a radius of `3`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

no overlap , ≈ 1.083

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)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (6 ,2) and (5 ,-4)

let `(x_1,y_1)=(6,2)" and " (x_2,y_2)=(5,-4)`

`d=sqrt((5-6)^2+(-4-2)^2)=sqrt(1+36)=sqrt37≈6.083`

sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii < d , then no overlap

smallest distance between them = d - sum of radii

`=6.083-5=1.083`
graph{(y^2-4y+x^2-12x+36)(y^2+8y+x^2-10x+32)=0 [-14.24, 14.24, -7.12, 7.12]}