Question

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

Answer 1

no overlap, ≈ 2.403

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_2)^2+(y_2-y_1)^2))color(white)(a/a)|)))`
where `(x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"`

here the 2 points are (2 ,-1) and (-3 ,3) the centres of the circles.

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

`d=sqrt((-3-2)^2+(3+1)^2)=sqrt(25+16)=sqrt41≈6.403`

sum of radii = radius of A + radius of B = 1 +3 = 4

Since sum of radii < d , then no overlap

min. distance = d - sum of radii = 6.403 - 4 = 2,403
graph{(y^2+2y+x^2-4x+4)(y^2-6y+x^2+6x+9)=0 [-10, 10, -5, 5]}