Question

Circle A has a radius of `2` and a center of `(6 ,6 )`. Circle B has a radius of `3` and a center of `(2 ,4 )`. If circle B is translated by `<1 ,5 >`, 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 it's position.

`"under the translation "((1),(5))`

`(2,4)to(2+1,4+5)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)|)))`

`"let "(x_1,y_1)=(6,6)" and "(x_2,y_2)=(3,9)`

`d=sqrt((3-6)^2+(9-6)^2)=sqrt18~~ 4.243`

`"sum of radii "=2+3=5`

`"since sum of radii">d" then circles overlap"`
graph{((x-6)^2+(y-6)^2-4)((x-3)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}