Question

Circle A has a radius of `2` and a center of `(2 ,5 )`. Circle B has a radius of `3` and a center of `(3 ,8 )`. If circle B is translated by `<4 ,-1 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap ",~~0.385`

Explanation:

`"what we have to do here is compare the distance (d)"`
`"between the centres to the 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"`

`"under the translation "<4,-1>`

`(3,8)to(3+4,8-1)to(7,7)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)=(2,5)" and "(x_2,y_2)=(7,7)`

`d=sqrt((7-2)^2+(7-5)^2)=sqrt(25+4)=sqrt29~~5.385`

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

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

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

`color(white)(xxxxxxxxxxxxx)=5.385-5=0.385`
graph{((x-2)^2+(y-5)^2-4)((x-7)^2+(y-7)^2-9)=0 [-20, 20, -10, 10]}