Question

Circle A has a radius of `2` and a center at `(5 ,1 )`. Circle B has a radius of `1` and a center at `(3 ,4 )`. If circle B is translated by `<-2 ,6 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 6.849

Explanation:

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

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Firstly ,we require to find the new centre of B under the translation.
A translation does not change the shape of a figure only it's position.

Under a translation of `((-2),(6))`

centre B(3 ,4) → (3-2 ,4+6) → (1 ,10)

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"`

let `(x_1,y_1)=(5,1)" and (x_2,y_2)=(1,10)`

`d=sqrt((1-5)^2+(10-1)^2)=sqrt(16+81)=sqrt97 ≈ 9.849`

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

Since sum of radii < d , then no overlap

min. distance between points = 9.849 - 3 = 6.849
graph{(y^2-2y+x^2-10x+22)(y^2-20y+x^2-2x+100)=0 [-40, 40, -20, 20]}