Question

Circle A has a radius of `6` and a center of `(2 ,5 )`. Circle B has a radius of `3` and a center of `(1 ,7 )`. If circle B is translated by `<3 ,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 compare the distance d "`
`"between the centres of the circles 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 "< 3,1>`

`(1,7)to(1+3,7+1)to(4,8)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)=(2,5)" and "(x_2,y_2)=(4,8)`

`d=sqrt((4-2)^2+(8-5)^2)=sqrt(4+9)=sqrt13~~3.61`

`"sum of radii "=6+3=9`

`"Since sum of radii">d" then circles overlap"`
graph{((x-2)^2+(y-5)^2-36)((x-4)^2+(y-8)^2-9)=0 [-40, 40, -20, 20]}