Question

Circle A has a radius of `2` and a center at `(8 ,3 )`. Circle B has a radius of `3` and a center at `(3 ,5 )`. 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 ",~~4.22`

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

`"under the translation "< -2,4>`

`(3,5)to(3-2,5+4)to(1,9)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)=(8,3)" and "(x_2,y_2)=(1,9)`

`d=sqrt((1-8)^2+(9-3)^2)=sqrt(49+36)=sqrt85~~9.22`

`"sum of radii "=2+3=5`

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

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

`color(white)(xxxxxxxxxx)=9.22-5=4.22`
graph{((x-8)^2+(y-3)^2-4)((x-1)^2+(y-9)^2-9)=0 [-40, 40, -20, 20]}