Question

Circle A has a radius of `3` and a center of `(2 ,5 )`. Circle B has a radius of `3` and a center of `(3 ,8 )`. If circle B is translated by `<4 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 1.07

Explanation:

What we have to do here is compare the distance (d) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

First we require to find the new position of centre B. A translation does not change the shape of a figure only it's position.

Under a translation of `((4),(2))`

centre of B(3 ,8) → (3+4 ,8+2) → (7 ,10)

To calculate d , 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)=(2,5)" and " (x_2,y_2)=(7,10)`

`d=sqrt((7-2)^2+(10-5)^2)=sqrt50 ≈ 7.07`

radius of A + radius of B = 3 + 3 = 6

Since sum of radii < d , then no overlap

and minimum distance = 7.06 - 6 = 1.06
graph{(y^2-10y+x^2-4x+20)(y^2-20y+x^2-14x+140)=0 [-35.56, 35.56, -17.78, 17.78]}