Question

Circle A has a radius of `2` and a center of `(7 ,6 )`. Circle B has a radius of `3` and a center of `(5 ,3 )`. If circle B is translated by `<-1 ,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 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

The first step is to find the new centre of B under the given translation. Under a translation the shape of the figure does not change only it's position.

Under a translation `((-1),(2))`

centre of B (5 ,3) → (5-1 ,3+2) → (4 ,5)

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 (7 ,6) and (4 ,5)

`d=sqrt((4-7)^2+(5-6)^2)=sqrt10≈3.162`

radius of A + radius of B = 2 + 3 = 5

Since sum of radii > d , then circles overlap.
graph{(y^2-12y+x^2-14x+81)(y^2-10y+x^2-8x+32)=0 [-20, 20, -10, 10]}