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 ,2 )`. If circle B is translated by `<-2 ,6 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap ",~~3.6`

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 "< -2,6>`

`(3,2)to(3-2,2+6)to(1,8)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)=(1,8)`

`d=sqrt((1-8)^2+(8-3)^2)`

`color(white)(d)=sqrt(49+25)=sqrt74~~8.60`

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

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

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

`color(white)(xxxxxxxxxxxxxx)=8.6-5=3.6`
graph{((x-8)^2+(y-3)^2-4)((x-1)^2+(y-8)^2-9)=0 [-20, 20, -10, 10]}