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 ,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 to the `color(blue)"sum of radii"`

`• " if "a>d" then circles overlap"`

`• " if "a < d" then no overlap"`

`"before we can calculate d we require to find the new "`
`"centre of B under the given translation"`

`"under the translation "<2,1>`

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

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

`color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(2/2)|)))`

`"let "(x_1,y_1)=(1,2)" and "(x_2,y_2)=(5,5)`

`d=sqrt((5-1)^2+(5-2)^2)=sqrt(16+9)=5`

`"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-5)^2-25)=0 [-20, 20, -10, 10]}