Question

Circle A has a center at `(8 ,3 )` and a radius of `1`. Circle B has a center at `(4 ,4 )` and a radius of `2`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

no overlap, ≈ 1.123

Explanation:

What we have to do here is compare the distance (d ) between the centres of the circles to the sum of their 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 points"`

the 2 points here are (8 ,3) and (4 ,4) the centres of the circles.

let `(x_1,y_1)=(8,3)" and " (x_2,y_2)=(4,4)`

`d=sqrt((4-8)^2+(4-3)^2)=sqrt(16+1)=sqrt17≈4.123`

sum of radii = radius of A + radius of B = 1 + 2 = 3

Since sum of radii < d , then no overlap

smallest distance = d - sum of radii = 4.123 - 3 = 1.123
graph{(y^2-6y+x^2-16x+72)(y^2-8y+x^2-8x+28)=0 [-10, 10, -5, 5]}