Question

Circle A has a radius of `2` and a center of `(3 ,1 )`. Circle B has a radius of `4` and a center of `(8 ,5 )`. If circle B is translated by `<-4 ,-1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"circles overlap"`

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 its position.

`"under a translation "((-4),(-1))`

`(8,5)to(8-4,5-1)to(4,4)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)|)))`

`(x_1,y_1)=(3,1)" and " (x_2,y_2)=(4,4)`

`d=sqrt((4-3)^2+(4-1)^2)=sqrt(1+9)=sqrt10~~3.162`

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

`"since sum of radii ">d" then circles overlap"`
graph{((x-3)^2+(y-1)^2-4)((x-4)^2+(y-4)^2-16)=0 [-10, 10, -5, 5]}