Question

Circle A has a center at `(-1 ,-4 )` and a radius of `3`. Circle B has a center at `(-3 ,2 )` and a radius of `2`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap , ≈ 1.325

Explanation:

What we have to do here is 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)(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."`

The 2 points here are (-1 ,-4) and (-3 ,2)

let `(x_1,y_1)=(-1,-4)" and " (x_2,y_2)=(-3,2)`

`d=sqrt((-3-(-1))^2+(2-(-4))^2)=sqrt(4+36)≈6.325`

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

Since sum of radii < d , then no overlap of circles.

smallest distance between them = d - sum of radii

`=6.325-5=1.325`
graph{(y^2-4y+x^2+6x+9)(y^2+8y+x^2+2x+8)=0 [-16.22, 16.26, -8.15, 8.08]}