Find the equation of a sphere which passes through `(1,0,0)(0,1,0)a n d(0,0,1),` and has radius as small as possible.

Explanation

Let the equation of the required sphere be `x^(2) +y^(2)+z^(2)+2ux+2vy+2wz+d=0" "` (i)
As the sphere passes through (1, 0, 0), (0, 1, 0) and (0, 0, 1), we get
`" "1+2u+d=0, 1+2v+d=0 and 1+2w+d=0`
`rArr" "u=v=w=-(1)/(2)(d+1)`
If R is the radius of the sphere, then
`" "R^(2)=u^(2)+v^(2)+w^(2)-d`
`" "=(3)/(4)(d+1)^(2)-d`
`" "=(3)/(4)[d^(2)+2d+1-(4)/(3)d]`
`" "=(3)/(4)[d^(2)+(2)/(3)d+1]`
`" "(3)/(4)[(d+(1)/(3))^(2)+1-(1)/(9)]`
`" "=(3)/(4)[(d+(1)/(3))^(2)+(8)/(9)]`
The last equation shows that `R^(2)` (and thus `R`) will be the least if and only if `d=-1//3`. Therefore,
`" "u=v=w=-(1)/(2)(1-(1)/(3))=-(1)/(3)`
Hence, the equation of the required sphere is `x^(2)+y^(2)+z^(2)-(2)/(3)(x+y+z)-(1)/(3)=0`
`" "3(x^(2)+y^(2)+z^(2))-2(x+y+z)-1=0`