Consider a sphere $S$ passing through the origin and the points $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. Find the radius of the sphere.

Solution

1. Let the sphere be $x^2+y^2+z^2+2gx+2fy+2hz+d=0$.
2. Passing through $(0,0,0)$ implies $d=0$.
3. Passing through $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ implies $1 + 2g = 0$, $1 + 2f = 0$, and $1 + 2h = 0$.
4. Thus $$\begin{array}{rl}g& =f\\ & =h\\ & =-1/2\end{array}$$, and the center is $(1/2,1/2,1/2)$.
5. The radius is $$\begin{array}{rl}\sqrt{(1/2{)}^2+(1/2{)}^2+(1/2{)}^2}& =\sqrt{3/4}\\ & =\frac{\sqrt{3}}{2}\end{array}$$.
6. Hence the answer is (A).