Question

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

Answer 1

`"no overlap ",~~1.39" to 2 dec. places"`

Explanation:

`"What we have to do here is compare the distance (d) "`
`"between the centres to the sum of their radii"`

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

`• " if sum of radii">d" then no overlap"`

`"Before we calculate d we require to find the centre of"`
`"B under the given translation"`

`"under the translation "<2,-3>`

`(4,7)to(4+2,7-3)to(6,4)larrcolor(red)"new centre of B"`

`"to calculate d use the "color(blue)"distance formula"`

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(6,4)" and "(x_2,y_2)=(1,2)`

`d=sqrt((1-6)^2+(2-4)^2)=sqrt(25+4)=sqrt29~~5.39`

`"sum of radii "=3+1=4`

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

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

`color(white)("minimum distance ")=5.39-4=1.39`
graph{((x-1)^2+(y-2)^2-9)((x-6)^2+(y-4)^2-1)=0 [-10, 10, -5, 5]}