Question

Circle A has a radius of `1` and a center of `(1 ,2 )`. Circle B has a radius of `2` and a center of `(8 ,1 )`. If circle B is translated by `<-4 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , minimum distance ≈ 0.61

Explanation:

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

• If the sum of radii > d , then circles overlap

• If the sum of radii < d , then no overlap

Before calculating d , we require to find the new centre of B under the translation.

Under a translation `((-4),(3))`

B(8 ,1) → B(8-4 ,1+3) → B(4 ,4)

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 coordinate points"`

The 2 points here are A(1 ,2) and B(4 ,4)

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

`d=sqrt((4-1)^2+(4-2)^2)=sqrt(9+4)=sqrt13≈3.61`

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

Since sum of radii < d , then no overlap

Minimum distance between 2 points on circles is.

d - sum of radii = 3.61 - 3 = 0.61
graph{(y^2-4y+x^2-2x+4)(y^2-8y+x^2-8x+28)=0 [-12.65, 12.66, -6.33, 6.32]}