Question

Circle A has a center at `(2 ,5 )` and a radius of `3`. Circle B has a center at `(7 ,2 )` and a radius of `3`. 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 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

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 (2 ,5) and (7 ,2) the centres of the circles.

let `(x_1,y_1)=(2,5)" and " (x_2,y_2)=(7,2)`

`d=sqrt((7-2)^2+(2-5)^2)=sqrt(25+9)=sqrt34≈5.831`

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

Since sum of radii > d , then circles overlap
graph{(y^2-10y+x^2-4x+20)(y^2-4y+x^2-14x+44)=0 [-24.98, 24.97, -12.5, 12.46]}