Question

Circle A has a radius of `4` and a center of `(6 ,1 )`. Circle B has a radius of `2` and a center of `(5 ,3 )`. 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

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

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

`(5,3)to(5-2,3+2)to(3,5)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)=(6,1)" and "(x_2,y_2)=(3,5)`

`d=sqrt((3-6)^2+(5-1)^2)=sqrt(9+16)=sqrt25=5`

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

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