Question

Circle A has a radius of `1` and a center of `(8 ,2 )`. Circle B has a radius of `4` and a center of `(5 ,3 )`. 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, min. distance ≈ 2.81

Explanation:

What we have to do here is `color(blue)"compare"` the distance (d ) between the centres of the circles to the `color(blue)"sum of the radii".`

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before calculating d, we require to find the 'new ' centre of B under the given translation, which does not change the shape of the circle, only it's position.

Under the translation `((-2),(5))`

`(5,3)to(5-2,3+5)to(3,8)larr" new centre of B"`

To calculate d, use the `color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

The 2 points here are (8 ,2) and (3 ,8)

let `(x_1,y_1)=(8,2)" and " (x_2,y_2)=(3,8)`

`d=sqrt((3-8)^2+(8-2)^2)=sqrt(25+36)=sqrt61≈7.81`

sum of radii = radius of A + radius of B = 1 + 4 = 5

Since sum of radii < d , then there is no overlap

min. distance between points = d - sum of radii

`=7.81-5=2.81`
graph{(y^2-4y+x^2-16x+67)(y^2-16y+x^2-6x+57)=0 [-32.05, 32.03, -16, 16.04]}