Find the equation of the sphere which passes through `(1,0,0),(0,1,0)a n d(0,0,1)` and whose center lies on the plane `3x-y+z=2.`

Explanation

Let the equation of the sphere be `x^(2)+y^(2)+z^(2)+2ux+2vy+2wz+d=0`.
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=-(d+1)/(2)`
Since the centre `(-u, -v, -w)` lies on the plane `3x-y+z=2`, we get `-3u+v-w=2`
`rArr" "(3)/(2)(d+1)=2 or d+1=(4)/(3) or d=(1)/(3)`
Thus, `u=v=w=-2//3`
Therefore, the equation of the required sphere is `x^(2)+y^(2)+z^(2)-((2)/(3))x-((2)/(3))y-((2)/(3))z+(1)/(3)=0`
or `" "3(x^(2)+y^(2)+z^(2))-2(x+y+z)+1=0`