Question

Circle A has a radius of `3` and a center of `(2 ,6 )`. Circle B has a radius of `4` and a center of `(7 ,3 )`. If circle B is translated by `<-3 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles overlap

Explanation:

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

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

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

Firstly we require to find the coordinates of the centre of B under the given translation.

Under a translation of `((-3),(2))`

centre B (7 ,3) → (7-3 ,3+2) → (4 ,5)

To calculate the distance (d) 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)" and " (x_2,y_2)" are 2 coord points"`

let `(x_1,y_1)=(2,6)" and " (x_2,y_2)=(4,5)`

`d=sqrt((4-2)^2+(5-6)^2)=sqrt(4+1)=sqrt5 ≈ 2.236`

radius of A + radius of B = 3 + 4 = 7

Since sum of radii > d , then circles overlap
graph{(y^2-12y+x^2-4x+31)(y^2-10y+x^2-8x+25)=0 [-15.59, 15.59, -7.8, 7.79]}