Question

Circle A has a radius of `3` and a center of `(3 ,2 )`. Circle B has a radius of `5` and a center of `(4 ,7 )`. If circle B is translated by `<2 ,-1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

circles overlap.

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),(-1))`

`(4,7)to(4+2,7-1)to(6,6)larr" 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"`

The 2 points here are (3 ,2) and (6 ,6)

let `(x_1,y_1)=(3,2)" and " (x_2,y_2)=(6,6)`

`d=sqrt((6-3)^2+(6-2)^2)=sqrt(9+16)=sqrt25=5`

Sum of radii = radius of A + radius of B =3 + 5 =8

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-6x+4)(y^2-12y+x^2-12x+47)=0 [-22.5, 22.5, -11.25, 11.25]}