Question

Circle A has a radius of `2` and a center of `(6 ,5 )`. Circle B has a radius of `1` and a center of `(3 ,4 )`. If circle B is translated by `<1 ,3 >`, 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 a translation "< 1,3 >`

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

`d=sqrt((4-6)^2+(7-5)^2)=sqrt(4+4)=sqrt8~~2.83`

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

`"Since sum of radii">d" then circles overlap"`
graph{((x-6)^2+(y-5)^2-4)((x-4)^2+(y-7)^2-1)=0 [-40, 40, -20, 20]}