Question

Circle A has a center at `(1 ,7 )` and a radius of `1`. Circle B has a center at `(-3 ,-2 )` and a radius of `2`. Do the circles overlap? If not, what is the smallest distance between them?

Answer 1

`"no overlap ",~~ 6.849`

Explanation:

what we have to do here is `color(blue)"compare "` the distance between the centres of the circles to the `color(blue)"sum of 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)|)))`

`(x_1,y_1)=(1,7)" and "(x_2,y_2)=(-3,-2)`

`d=sqrt((-3-1)^2+(-2-7)^2)=sqrt(16+81)~~ 9.849`

`"sum of radii "=1+2=3`

`"since sum of radii"< d" then no overlap"`

`"smallest distance "=d-" sum of radii"`

`color(white)(smallest distance)=9.849-3=6.849`
graph{((x-1)^2+(y-7)^2-1)((x+3)^2+(y+2)^2-4)=0 [-20, 20, -10, 10]}