Question

Circle A has a radius of `4` and a center of `(7 ,3 )`. Circle B has a radius of `2` and a center of `(1 ,2 )`. If circle B is translated by `<2 ,3 >`, 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 calculate the distance (d) between the centres of A and B and compare this with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

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

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

Calculate d using 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)=(7,3)" and " (x_2,y_2)=(3,5)`

`d=sqrt((3-7)^2+(5-3)^2)=sqrt(16+4)=sqrt20 ≈ 4.472`

radius of A + radius of B = 4 + 2 = 6

Since sum of radii > d , circles overlap