Question

Circle A has a center at `(7 ,-2 )` and a radius of `2`. Circle B has a center at `(4 ,2 )` and a radius of `4`. 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 to 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)=(7,-2)" and " (x_2,y_2)=(4,2)`

`d=sqrt((4-7)^2+(2+2)^2)=sqrt(9+16)=sqrt25=5`

radius of A + radius of B = 2 + 4 = 6

Since sum of radii > d , then circles overlap
graph{(y^2+4y+x^2-14x+49)(y^2-4y+x^2-8x+4)=0 [-11.25, 11.25, -5.625, 5.625]}