Question

Circle A has a center at `(-1 ,3 )` and a radius of `5`. Circle B has a center at `(3 ,-2 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

`"no overlap, smallest distance" ≈0.403`

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 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)(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 (-1 ,3) and (3 ,-2)

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

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

Sum of radii = radius of A + radius of B = 5 + 1 = 6

Since sum of radii < d , then no overlap

smallest distance = d - sum of radii

`=6.403-6=0.403`
graph{(y^2-6y+x^2+2x-15)(y^2+4y+x^2-6x+12)=0 [-20, 20, -10, 10]}