Question

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

Answer 1

no overlap,≈ 1.657

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 (3 ,5) and (-1 ,1)

let `(x_1,y_1)=(3,5)" and " (x_2,y_2)=(-1,1)`

`d=sqrt((-1-3)^2+(1-5)^2)=sqrt(16+16)=sqrt32≈5.657`

sum of radii = radius of A + radius of B = 1 + 3 = 4

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

smallest distance between them = d - sum of radii

`=5.657-4=1.657" units"`
graph{(y^2-10y+x^2-6x+33)(y^2-2y+x^2+2x-7)=0 [-20, 20, -10, 10]}