Question

Circle A has a center at `(4 ,-1 )` and a radius of `3`. Circle B has a center at `(-3 ,6 )` and a radius of `2`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap, min. distance ≈ 4.899

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 ,-1) and (-3 ,6) the centres of the circles.

let `(x_1,y_1)=(4,-1)" and " (x_2,y_2)=(-3,6)`

`d=sqrt((-3-4)^2+(6+1)^2)=sqrt(49+49)=sqrt98≈9.899`

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

Since sum of radii < d , then no overlap

smallest distance = d - sum of radii = 9.899 - 5 = 4.899
graph{(y^2+2y+x^2-8x+8)(y^2-12y+x^2+6x+41)=0 [-20, 20, -10, 10]}