Question

Circle A has a radius of `4` and a center of `(8 ,5 )`. Circle B has a radius of `3` and a center of `(6 ,7 )`. If circle B is translated by `<3 ,1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles overlap

Explanation:

To determine wether the circles overlap or not , requires calculating the distance (d) between the centres and comparing 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 `((3),(1))`

centre of B(6 , 7) → (6 + 3 , 7 + 1) → (9 , 8)

To calculate the distance (d) 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 coordinate points "`

let `(x_1,y_1)=(8,5)" and " (x_2,y_2)=(9,8)`

`rArr d =sqrt((9-8)^2+(8-5)^2)=sqrt(1+9)=sqrt10 ≈ 3.16`

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

Since sum of radii > d , then circles overlap.