The equation of the sphere passing through the origin and the points $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ is:

Solution

1. Since it passes through the origin, $d=0$.
2. Passing through $(1,0,0)$ gives $1 + 2u = 0$, so $u=-1/2$.
3. Passing through $(0,1,0)$ gives $1 + 2v = 0$, so $v=-1/2$.
4. Passing through $(0,0,1)$ gives $1 + 2w = 0$, so $w=-1/2$.
5. The equation is $x^2+y^2+z^2-x-y-z=0$.
6. Hence the answer is (A).