Question

Circle A has a radius of `2` and a center of `(5 ,7 )`. Circle B has a radius of `4` and a center of `(7 ,2 )`. If circle B is translated by `<2 ,-5 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap, ≈ 4.77

Explanation:

To determine this , we require to compare the distance (d) between the centres of the circles with the sum of their radii.

• If sum of radii > d , then circles will overlap.

• If sum of radii < d , then no overlap will occur.

A translation does not change the shape of a figure , only it's position.

Under a translation of `((2),(-5))`

centre of B (7 , 2) → (7+2 ,2-5 ) → (9 , -3)

To calculate the distance between the 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)=(5 , 7)" and " (x_2,y_2)=(9 , -3)`

`rArr d = sqrt((9-5)^2 + (-3-7)^2) =sqrt(16+100) ≈ 10.77`

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

Since sum of radii < d , no overlap

and distance between circles = 10.77 - 6 = 4.77