Question

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

Answer 1

circle A enclosed inside circle B

Explanation:

What we have to do here is compare the distance (d) between the centres with the sum and difference of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

• If difference of radii > d , then 1 circle inside other

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 the centres (3 ,2) and (1 ,3)

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

`d=sqrt((3-2)^2+(1-3)^2)=sqrt(1+4)=sqrt5≈2.236`

sum of radii = radius of A + radius of B = 1 + 4 = 5

difference of radii = radius of B - radius of A = 4 - 1 = 3

Since difference of radii > d , then circle A is enclosed inside circle B.
graph{(y^2-4y+x^2-6x+12)(y^2-6y+x^2-2x-6)=0 [-10, 10, -5, 5]}