Question

Circle A has a radius of `5` and a center of `(8 ,2 )`. Circle B has a radius of `3` and a center of `(3 ,7 )`. If circle B is translated by `<4 , 8 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap "~~5.01`

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 "< 4,8>`

`(3,7)to(3+4,7+8)to(7,15)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)=(8,2)" and "(x_2,y_2)=(7,15)`

`d=sqrt((7-8)^2+(15-2)^2)=sqrt(1+169)=sqrt170~~13.01`

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

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

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

`color(white)(xxxxxxxxxx)=13.01-8=5.01`
graph{((x-8)^2+(y-2)^2-25)((x-7)^2+(y-15)^2-9)=0 [-40, 40, -20, 20]}