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 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~2.82`

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 "<2,4>`

`(3,7)to(3+2,7+4)to(5,11)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)=(5,11)`

`d=sqrt((5-1)^2+(11-2)^2)=sqrt(16+81)=sqrt117~~10.82`

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

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

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

`color(white)(xxxxxxxxxxxxx)=10.82-8=2.82`
graph{((x-1)^2+(y-2)^2-9)((x-5)^2+(y-11)^2-25)=0 [-20, 20, -10, 10]}