Question

Circle A has a radius of `1` and a center of `(1 ,7 )`. Circle B has a radius of `2` and a center of `(8 ,1 )`. If circle B is translated by `<-4 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~1.24`

Explanation:

`"what we have to do here is "color(blue)"compare"'"the distance (d)"`
`"between the centres to the "color(blue)"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 centre of B"`
`"under the given translation"`

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

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

`d=sqrt((4-1)^2+(4-7)^2)=sqrt(9+9)=sqrt18~~4.24`

`"sum of radii "=1+2=3`

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

`"min. distance "=d-" sum of radii"`

`color(white)(xxxxxxxxxx)=4.24-3=1.24`
graph{((x-1)^2+(y-7)^2-1)((x-4)^2+(y-4)^2-4)=0 [-20, 20, -10, 10]}