Question

Circle A has a center at `(5 ,-2 )` and a radius of `2`. Circle B has a center at `(1 ,-4 )` and a radius of `1`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

no overlap, smallest distance ≈ 1.47 units

Explanation:

What we have to do here is `color(blue)"compare"` the distance (d) between the centres of the circles to the `color(blue)"sum of 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)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 points here are (5 ,-2) and (1 ,-4)

let `(x_1,y_1)=(5,-2)" and " (x_2,y_2)=(1,-4)`

`d=sqrt((1-5)^2+(-4+2)^2)=sqrt(16+4)=sqrt20≈4.47`

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

Since sum of radii < d, then no overlap of circles

smallest distance between circles = d - sum of radii

`=4.47-3=1.47`
graph{(y^2+4y+x^2-10x+25)(y^2+8y+x^2-2x+16)=0 [-10, 10, -5, 5]}