Question

Circle A has a radius of `4` and a center of `(6 ,2 )`. Circle B has a radius of `2` and a center of `(5 ,7 )`. If circle B is translated by `<-2 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap ",~~1.616`

Explanation:

What we have to do here is `color(blue)"compare"` the distance ( d )between the centres of the circles to the `color(blue)"sum of 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"` which does not change the shape of the circle only it's position.

`"Under a translation " ((-2),(2))`

`(5,7)to(5-2,7+2)to(3,9)larrcolor(red)" new centre of B"`

`"to calculate d use 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)|)))`

`"where " (x_1,y_1),(x_2,y_2)" are 2 coordinate points"`

`"the 2 points here are " (6,2)" and " (3,9)`

`d=sqrt((3-6)^2+(9-2)^2)=sqrt(9+49)=sqrt58~~7.616`

`"sum of radii " =4+2=6`

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

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

`rArr"min distance "=7.616-6=1.616`
graph{(y^2-4y+x^2-12x+24)(y^2-18y+x^2-6x+86)=0 [-28.87, 28.86, -14.43, 14.44]}