Question

Circle A has a radius of `5` and a center of `(6 ,1 )`. Circle B has a radius of `1` and a center of `(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"`

Explanation:

What we have to do here is `color(blue)"compare "` thedistance (d) between the centres with the `color(blue)"sum of radii"`

`• " if sum of radii">d" then circles overlap"`

`• " if sum of radii"< d" then no overlap"`

`"before we can calculate d we require the 'new' centre of"`
`"circle B"`

`"under a translation of "<-3,4>`

`(4,5)to(4-3,5+4)to(1,9)larrcolor(red)"new centre of B"`

`"calculate d using the "color(blue)"distance formula"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`

`"let "(x_1,y_1)=(6,1)" and "(x_2,y_2)=(1,9)`

`d=sqrt((1-6)^2+(9-1)^2)=sqrt(25+64)~~9.43`

`"sum of radii "=5+1=6`

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

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

`color(white)("minimum distance ")=9.43-6=3.43`
graph{((x-6)^2+(y-1)^2-25)((x-1)^2+(y-9)^2-1)=0 [-20, 20, -10, 10]}