Prove that all points on the general cone with vertex $(x_0,y_0,z_0)$ and generator $y(s)$ are elliptic or parabolic points.
I solved this problem somehow, but I got that all the points are elliptic so i got confused. Firstly I'm not sure if I have chosen the right equation of the general cone with given vertex and generator, I took the parametric equation:
$x=(a*u*cos(v),a*u*sin(v),u)$
From this equation I found the coefficients of the second fundamental form L,M,N
$L=0$
$N=0$
$M=\frac{a*u}{\sqrt{1+a^2}}$
So $L*N-M^2=\frac{a^2*u^2}{1+a^2}>0$ it means the points are elliptic. But I dont know how to prove that they might be parabolic too.
Or should I take the other equation for the general cone as $x=y(s)+u*g(s)$ where $g$ is a constant vector.Howerver I'm not sure if this equation is true.
thank you all for your help!