Question

Circle A has a radius of `2` and a center of `(7 ,2 )`. Circle B has a radius of `3` and a center of `(5 ,7 )`. If circle B is translated by `<-1 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 2.616 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 no overlap

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

Under a translation `((-1),(2))`

`B(5,7)to(5-1,7+2)to(4,9)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)" and " (x_2,y_2)" are 2 coordinate points"`

The 2 points here are (7 ,2) and (4 ,9)

let `(x_1,y_1)=(7,2)" and " (x_2,y_2)=(4,9)`

`d=sqrt((4-7)^2+(9-2)^2)=sqrt(9+49)=sqrt58≈7.616`

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

Since sum of radii < d, then no overlap

and min. distance between points = d - sum of radii

`=7.616-5=2.616`
graph{(y^2-4y+x^2-14x+49)(y^2-18y+x^2-8x+88)=0 [-25.31, 25.32, -12.66, 12.65]}