Question

Circle A has a center at `(2 ,4 )` and a radius of `5`. Circle B has a center at `(9 ,3 )` and a radius of `1`. Do the circles overlap? If not what is the smallest distance between them?

Answer 1

`"no overlap ",~~1.07`

Explanation:

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

`"let "(x_1,y_1)=(2,4)" and "(x_2,y_2)=(9,3)`

`d=sqrt((9-2)^2+(3-4)^2)=sqrt(49+1)=sqrt50~~7.07`

`"sum of radii "=5+1=6`

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

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

`color(white)(xxxxxxxxxxxx)=7.07-6=1.07`
graph{((x-2)^2+(y-4)^2-25)((x-9)^2+(y-3)^2-1)=0 [-20, 20, -10, 10]}