Question

Circle A has a radius of `4` and a center of `(6 ,2 )`. Circle B has a radius of `3` and a center of `(5 ,7 )`. If circle B is translated by `<-2 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~0.62`

Explanation:

`"what we have to do here is compare the distance (d)"`
`"between the centres to the sum of the radii"`

`• " if sum of radii">d" then circles overlap"`

`• " if sum of radii"< d " then no overlap"`

`"before calculating d we require to find the new centre"`
`"of B under the given translation"`

`"under a translation "< -2,2>`

`(5,7)to(5-2,7+2)to(3,9)larrcolor(red)"new centre of B"`

`"to calculate d use the "color(blue)"distance formula"`

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(6,2)" and "(x_2,y_2)=(3,9)`

`d=sqrt((3-6)^2+(9-2)^2)=sqrt(9+49)=sqrt58~~7.62`

`"sum of radii "=4+3=7`

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

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

`color(white)(xxxxxxxxxxxxx)=7.62-7=0.62`
graph{((x-6)^2+(y-2)^2-16)((x-3)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}