Question

Circle A has a radius of `4` and a center of `(6 ,1 )`. Circle B has a radius of `1` and a center of `(5 ,3 )`. If circle B is translated by `<-2 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles have a single point of contact.

Explanation:

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

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

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

Now require to calculate the distance between the centres of A and B 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 coordinate points "`

let `(x_1,y_1)=(6,1)" and " (x_2,y_2)=(3,5)`

d `= sqrt((3-6)^2 + (5-1)^2) = sqrt(9 + 16) = sqrt25 = 5`

now: radius of A + radius of B = 4 + 1 = 5

Since sum of radii = distance between centres , then the circles will have a single point of contact.