Question

Circle A has a radius of `3` and a center of `(2 ,4 )`. Circle B has a radius of `2` and a center of `(4 ,7 )`. If circle B is translated by `<2 ,-4 >`, 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 "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"`

`"we require to find the new centre of B under the given"`
`"translation"`

`(4,7)to(4+2,7-4)to(6,3)larrcolor(red)"new centre of B"`

`"to find 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)=(2,4)" and "(x_2,y_2)=(6,3)`

`d=sqrt((6-2)^2+(3-4)^2)=sqrt(16+1)=sqrt17~~4.123`

`"sum of radii "=3+2=5`

`"since sum of radii">d" then circles overlap"`
graph{((x-2)^2+(y-4)^2-9)((x-6)^2+(y-3)^2-4)=0 [-20, 20, -10, 10]}