Question

Circle A has a radius of `1` and a center at `(7 ,4 )`. Circle B has a radius of `3` and a center at `(6 ,5 )`. If circle B is translated by `<-3 ,4 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~2.403`

Explanation:

What we have to do here is `color(blue)"compare"` the distance between the centres of the circles with the `color(blue)"sum of radii"`

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

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

`"we require to find the new centre of circle B under"`
`"the given translation which does not change the shape"`
`"of the circle only it's position"`

`"under a translation "<-3,4>`

`(6,5)to(6-3,5+4)to(3,9)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)=(7,4)" and "(x_2,y_2)=(3,9)`

`d=sqrt((3-7)^2+(9-4)^2)=sqrt41~~6.403`

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

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

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

`color(white)(xxxxxxxxxxxxxx)=6.403-4=2.403`
graph{((x-7)^2+(y-4)^2-1)((x-3)^2+(y-9)^2-9)=0 [-20, 20, -10, 10]}