Question

Circle A has a radius of `4` and a center at `(8 ,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 "~~2.9`

Explanation:

`"What we have to do here is to compare the distance (d)"`
`"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 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)=(8,2)" and "(x_2,y_2)=(1,9)`

`d=sqrt((1-8)^2+(9-2)^2)=sqrt(49+49)=sqrt98~~9.9`

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

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

`"minimum distance "=d-" sum of radii"`

`color(white)(xxxxxxxxxxxxx)=9.9-7=2.9`
graph{((x-8)^2+(y-2)^2-16)((x-1)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}