Question

Circle A has a radius of `3` and a center of `(2 ,1 )`. 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 require here is to 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

The first step is to find the coordinates of B under the translation. A translation does not change the shape of the circle only it's position.

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

B(7 ,3) → (7-3 ,3+2) → B(4 ,5) is new position of centre B

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"`

Here the 2 points are A(2 ,1) and B(4 ,5)

let `(x_1,y_1)=(2,1)" and " (x_2,y_2)=(4,5)`

`d=sqrt((4-2)^2+(5-1)^2)=sqrt(4+16)=sqrt20≈4.472`

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

Since sum of radii > d , then circles overlap
graph{(y^2-2y+x^2-4x-5)(y^2-10y+x^2-8x+25)=0 [-20, 20, -10, 10]}