Question

Circle A has a radius of `3` and a center of `(5 ,4 )`. Circle B has a radius of `1` and a center of `(7 ,2 )`. If circle B is translated by `<3 ,-5 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

`"no overlap ",~~4.602`

Explanation:

`"what we have to do here is "color(blue)"compare"" the distance (d)"`
`"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 "<3,-5>`

`(7,2)to(7+3,2-5)to(10,-3)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)=(5,4)" and "(x_2,y_2)=(10,-3)`

`d=sqrt((10-5)^2+(-3-4)^2)=sqrt(25+49)~~8.062`

`"sum of radii "=3+1=4`

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

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

`color(white)(xxxxxxxxxx)=8.062-4=4.062`
graph{((x-5)^2+(y-4)^2-9)((x-10)^2+(y+3)^2-1)=0 [-20, 20, -10, 10]}