Question

Circle A has a center at `(5 ,2 )` and a radius of `4`. Circle B has a center at `(9 ,8 )` and a radius of `7`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

circles overlap

Explanation:

What we have to do here is compare the distance (d ) between the centres with the sum of the radii.

• If sum of radii > d , then circles overlap

• If sum of radii < d , then no overlap

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 points"`

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

`d=sqrt((9-5)^2+(8-2)^2)=sqrt(16+36)≈7.211`

radius of A + radius of B = 4 + 7 = 11

Since sum of radii > d , then circles overlap
graph{(y^2-4y+x^2-10x+13)(y^2-16y+x^2-18x+96)=0 [-40, 40, -20, 20]}