Question

Circle A has a radius of `4` and a center at `(7 ,2 )`. Circle B has a radius of `3` and a center at `(4 ,5 )`. If circle B is translated by `<-3 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap ",~~2.22`

Explanation:

`"what we have to do here is "color(blue)"compare ""the distance (d)"`
`"to the "color(blue)"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 "<-3,4>`

`(4,5)to(4-3,5+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)=(7,2)" and "(x_2,y_2)=(1,9)`

`d=sqrt((1-7)^2+(9-2)^2)=sqrt(36+49)~~9.22`

`"sum of radii "=4+3=7`

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

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

`color(white)("min. distance ")=9.22-7=2.22`
graph{((x-7)^2+(y-2)^2-16)((x-1)^2+(y-9)^2-9)=0 [-40, 40, -20, 20]}