Question

Circle A has a radius of `4` and a center of `(8 ,5 )`. Circle B has a radius of `2` and a center of `(6 ,1 )`. If circle B is translated by `<3 ,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 compare the distance (d )"`
`"between the centres of the circles to the"`
`color(blue)"sum of the radii"`

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

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

Before calculating d we require to find the coordinates of the new centre of B under the given translation which does not change the shape of the circle only its position.

`"under a translation "((3),(1))`

`(6,1)to(6+3,1+1)to(9,2)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)=(8,5),(x_2,y_2)=(9,2)`

`d=sqrt((9-8)^2+(2-5)^2)=sqrt(1+9)=sqrt10~~3.162`

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

`"since sum of radii > d then circles overlap"`
graph{(y^2-10y+x^2-16x+73)(y^2-4y+x^2-18x+81)=0 [-20, 20, -10, 10]}