Question

Circle A has a radius of `1` and a center of `(2 ,4 )`. Circle B has a radius of `2` and a center of `(4 ,9 )`. If circle B is translated by `<1 ,-4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap" ,~~0.162`

Explanation:

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

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

`(4,9)to(4+1,9-4)to(5,5)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 points are " (x_1,y_1)=(2,4),(x_2,y_2)=(5,5)`

`d=sqrt((5-2)^2+(5-4)^2)=sqrt10~~3.162`

`"sum of radii " =1+2=3`

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

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

`rArr"minimum distance " =3.162-3=0.162`
graph{(y^2-8y+x^2-4x+19)(y^2-10y+x^2-10x+46)=0 [-10, 10, -5, 5]}