Question

Circle A has a radius of `2` and a center at `(5 ,2 )`. Circle B has a radius of `5` and a center at `(3 ,4 )`. If circle B is translated by `<2 ,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 of circles"`

Before finding 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 it's position.

`" Under a trans;ation " ((2),(1))`

`(3,4)to(3+2,4+1)to(5,5)larrcolor(red)" new centre of B"`

Since the x-ccordinates of both centres are 5 then the centres lie on the same vertical line and d is the difference in their y-coordinates.

`rArrd=5-2=3`

`"sum of radii = radius of A + radius of B "=2+5=7`

`"since sum of radii"> d" then circles overlap"`
graph{(y^2-4y+x^2-10x+25)(y^2-10y+x^2-10x+25)=0 [-13.8, 13.91, -6.93, 6.92]}