How to identify any point inside or outside the given cone?

Question

The equation of a double circular cone with a vertex $p=(a,b,c)$ with the generating angle $t$ is given by

$(x-a)^2+(y-b)^2= \frac{(z-c)^2}{t^2}$

How do I identify the point $A=(x_{0},y_{0},z_{0})$ inside or outside the given cone?

Answer 1

Extension of the technique done in case of circles.

The cross section of the cone at $z=z_0$ is a circle, with center $(a,b,z_0)$, now check whether A is inside this circle i.e. check whether $C(a,b,z_0).C(x_0,y_0,z_0) > 0$.