Question

Circle A has a radius of `2` and a center at `(3 ,6 )`. Circle B has a radius of `4` and a center at `(2 ,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 new"`
`"centre of B under the given translation"`

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

`B(2,3)to(2-2,3+1)to(0,4)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)=(3,6)" and "(x_2,y_2)=(0,4)`

`d=sqrt((0-3)^2+(4-6)^2)=sqrt(9+4)=sqrt13~~3.61`

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

`"Since sum of radii"> d" then circles overlap"`
graph{((x-3)^2+(y-6)^2-4)((x-0)^2+(y-4)^2-16)=0 [-20, 20, -10, 10]}