Question

Circle A has a radius of `3` and a center of `(2 ,1 )`. Circle B has a radius of `2` and a center of `(7 ,8 )`. If circle B is translated by `<4 ,2 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap" ~~7.727`

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 "((4),(2))`

`(7,8)to(7+4,8+2)to(11,10)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)=(2,1)" and "(x_2,y_2)=(11,10)`

`d=sqrt((11-2)^2+(10-1)^2)=sqrt(162)~~ 12.727`

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

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

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

`color(white)(minimum distance)=12.727-5=7.727`
graph{((x-2)^2+(y-1)^2-9)((x-11)^2+(y-10)^2-4)=0 [-20, 20, -10, 10]}