Question

Circle A has a radius of `6` and a center of `(2 ,5 )`. Circle B has a radius of `3` and a center of `(6 ,7 )`. If circle B is translated by `<3 ,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 require to do here is compare the distance ( d) between the centres of the circles to the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

Before doing this we need 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 `((3),(1))`

(6 ,7) → (6+3 ,7+1) → (9 ,8) is the 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 (2 ,5) and (9 ,8) the centres of the circles.

let `(x_1,y_1)=(2,5)" and " (x_2,y_2)=(9,8)`

`d=sqrt((9-2)^2+(8-5)^2)=sqrt(49+9)=sqrt58≈7.616`

sum of radii = radius of A + radius of B = 6 + 3 = 9

Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x-7)(y^2-16y+x^2-18x+136)=0 [-40, 40, -20, 20]}