Question

Circle A has a radius of `3` and a center of `(3 ,2 )`. Circle B has a radius of `1` and a center of `(4 ,7 )`. 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

cicles overlap

Explanation:

What we have to do here is to calculate the distance (d) between the centres of the circles 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 (4 ,7) → (4+2 , 7-3) → (6 , 4)

To calculate the distance between 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,y_1)" and " (x_2,y_2)" are 2 points "`

let `(x_1,y_1)=(3,2)" and " (x_2,y_2)=(6,4)`

`d=sqrt((6-3)^2+(4-2)^2)=sqrt(9+4)=sqrt13 ≈ 3.61`

radius of A + radius of B = 3 + 1 = 4

Since sum of radii > d , then circles overlap