Show that if all normals to a connected surface pass through the origin, the surface is contained in a sphere.
The Attempt at a Solution
I know a surface is locally the graph of a differentiable function, so in a neighbourhood of a point p, the points satisfy the equation F(x,y,z) = 0. Then a normal vector would be grad(F)(p), and the parameterized normal line would be X(t) = p + t*grad(F)(p).
I know that line passes through the origin, so for some t, X(t)=(0,0,0). But then I am lost. I don't really know how to handle the problem.
I also thought of taking a parameterization of a neighbourhood of p, then a basis for the tangent plane is given by the partial derivatives of the parameterization, and a normal vector is the vector product of those derivatives.
Thanks for any help :)