Question

Circle A has a center at `(3 ,7 )` and a radius of `4`. Circle B has a center at `(4 ,-2 )` and a radius of `6`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

circles overlap.

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 (3 ,7) and (4 ,-2) the centres of the circles.

let `(x_1,y_1)=(3,7)" and " (x_2,y_2)=(4,-2)`

`d=sqrt((4-3)^2+(-2-7)^2)=sqrt(1+81)=sqrt82≈9.055`

sum of radii = radius of A + radius of B = 4 + 6 = 10

Since sum of radii > d , then circles overlap
graph{(y^2-14y+x^2-6x+42)(y^2+4y+x^2-8x-16)=0 [-40, 40, -20, 20]}