Question

Circle A has a center at `(-4 ,-3 )` and a radius of `2`. Circle B has a center at `(2 ,-1 )` and a radius of `3`. 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 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 (-4 ,-3) and (2 ,-1) the centres of the circles.

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

`d=sqrt((2+4)^2+(-1+3)^2)=sqrt(36+4)=sqrt40≈6.325`

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

Since sum of radii < d, then there is no overlap

min. distance between them = d - sum of radii

`=6.325 - 5 = 1.325`
graph{(y^2+6y+x^2+8x+21)(y^2+2y+x^2-4x-4)=0 [-10, 10, -5, 5]}