Question

Circle A has a center at `(1 ,4 )` and a radius of `5`. Circle B has a center at `(9 ,3 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

There is a minimum distance of `(sqrt(65)-6)~~2.06` units between the two (non-overlapping) circles.

Explanation:

The distance between the two centers is
`color(white)("XXX")d=sqrt((9-1)^2+(4-3)^2)=sqrt(65)~~8.06`

Along the line segment connecting the two centers
`5` units are covered by circle A, and
`1` unit is covered by circle B.

So only `5+1=6` units are covered by the circles.

Leaving `sqrt(65)-6~~2.06` units uncovered.

Answer 2

no overlap , ≈ 2.06

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

To calculate the distance between the centres 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"`

let `(x_1,y_1)=(1,4)" and " (x_2,y_2)=(9,3)`

`d=sqrt((9-1)^2+(3-4)^2)=sqrt65 ≈ 8.06`

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

Since sum of radii < d , then no overlap

smallest distance = 8.06 - 6 = 2.06
graph{(y^2-8y+x^2-2x-8)(y^2-6y+x^2-18x+89)=0 [-35.56, 35.56, -17.78, 17.78]}