Question

Circle A has a radius of `2` and a center of `(4 ,6 )`. Circle B has a radius of `3` and a center of `(9 ,8 )`. If circle B is translated by `<-2 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap, min. distance ≈ 1.708 units.

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 there is no overlap

Before calculating d we require to find the 'new' centre of B under the given translation which does no change the shape of the circle only it's position.

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

`(9,8)to(9-2,8+4)to(7,12)rarrcolor(red)" 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 (4 ,6) and (7 ,12)

let `(x_1,y_1)=(4,6)" and " (x_2,y_2)=(7,12)`

`d=sqrt((7-4)^2+(12-6)^2)=sqrt(9+36)=sqrt45~~6.708`

Sum of radii = radius of A + radius of B = 2 + 3 = 5

Since sum of radii < d, then no overlap

min. distance = d - sum of radii

`rArr" min. distance "=6.708-5=1.708`
graph{(y^2-12y+x^2-8x+48)(y^2-24y+x^2-14x+184)=0 [-40, 39.98, -20, 19.98]}