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 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~0.71" units"`

Explanation:

`"What we have to do here is to "color(blue)"compare ""the"`
`"distance (d) between the centres with the "color(blue)"sum of 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 the translation "<-2,4>`

`(8,3)to(8-2,3+4)to(6,7)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)=(3,1)" and "(x_2,y_2)=(6,7)`

`d=sqrt((6-3)^2+(7-1)^2)=sqrt(9+36)=sqrt45~~6.71`

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

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

`"min. distance "=d-" sum of radii"`

`color(white)("min. distance ")=6.71-6=0.71`
graph{((x-3)^2+(y-1)^2-4)((x-6)^2+(y-7)^2-16)=0 [-20, 20, -10, 10]}