Question

Circle A has a radius of `4` and a center at `(7 ,2 )`. Circle B has a radius of `3` and a center at `(6 ,5 )`. If circle B is translated by `<-3 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

no overlap , ≈ 1.062

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

However, the first step here is to calculate the new centre of B under the given translation, which does not change the shape of the circle only it's position.

Under a translation `((-3),(4))`

`B(6,5)to(6-3,5+4)to(3,9)" new centre of B"`

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 coordinate points"`

here the 2 points are (7 ,2) and (3 ,9) the centres of the circles.

let `(x_1,y_1)=(7,2)" and " (x_2,y_2)=(3,9)`

`d=sqrt((3-7)^2+(9-2)^2)=sqrt(16+49)=sqrt65≈8.062`

sum of radii = radius of A + radius of B = 4 + 3 = 7

Since sum of radii < d , then no overlap of circles

min. distance = d - sum of radii = 8.062 - 7 = 1.062
graph{(y^2-4y+x^2-14x+37)(y^2-18y+x^2-6x+81)=0 [-40, 40, -20, 20]}