Question

Circle A has a radius of `3` and a center at `(1 ,2 )`. Circle B has a radius of `5` and a center at `(3 ,7 )`. 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 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 "<2,1>`

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

`d=sqrt((5-1)^2+(8-2)^2)=sqrt(16+36)=sqrt52~~7.211`

`"sum of radii "=3+5=8`

`"since sum of radii"> d" then circles overlap"`
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-8)^2-25)=0 [-20, 20, -10, 10]}