Question

Circle A has a center at `(-5 ,8 )` and a radius of `4`. Circle B has a center at `(-3 ,3 )` 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 `color(blue)"compare"` the distance (d) between the centres of the circles with the `color(blue)"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)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`
where `(x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

`"the 2 points here are " (-5,8)" and " (-3,3)`

`"let " (x_1,y_1)=(-5,8)" and " (x_2,y_2)=(-3,3)`

`d=sqrt((-3+5)^2+(3-8)^2)=sqrt(4+25)=sqrt29≈5.385`

`"sum of radii = radius of A + radius of B = 4+4= 8"`

#Since sum of radii > d , then circles overlap.
graph{(y^2-16y+x^2+10x+73)(y^2-6y+x^2+6x+2)=0 [-28.48, 28.47, -14.24, 14.24]}