Question

Circle A has a center at `(-9 ,8 )` and a radius of `3`. Circle B has a center at `(-8 ,3 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

no overlap , ≈ 1.099

Explanation:

What we have to do here is compare the distance (d ) between the centre 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 (-9 ,8) and (-8 ,3) the centres of the circles.

let `(x_1,y_1)=(-9,8)" and " (x_2,y_2)=(-8,3)`

`d=sqrt((-8+9)^2+(3-8)^2)=sqrt(1+25)=sqrt26≈5.099`

sum of radii = radius of A + radius of B = 3 + 1 = 4

Since sum of radii < d , then no overlap

min. distance = d - sum of radii = 5.099 - 4 = 1.099
graph{(y^2-16y+x^2+18x+136)(y^2-6y+x^2+16x+72)=0 [-25.31, 25.32, -12.66, 12.65]}