Question

Circle A has a radius of `2` and a center at `(3 ,1 )`. Circle B has a radius of `4` and a center at `(8 ,3 )`. If circle B is translated by `<-2 ,1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"circles overlap"`

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 centre of B"`
`"under the given translation"`

`"under a translation "<-2,1>`

`(8,3)to(8-2,3+1)to(6,4)larrcolor(red)"new centre of B"`

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

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

`"let "(x_1,y_1)=(6,4)" and "(x_2,y_2)=(3,1)`

`d=sqrt((3-6)^2+(1-4)^2)=sqrt(9+9)=sqrt18~~4.24`

`"sum of radii "=2+4=6`

`"since sum of radii">d" then circles overlap"`
graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-4)^2-16)=0 [-10, 10, -5, 5]}