I have the following equation of a conic in trilinear coordinates:
$$x^2+y^2+z^2-\frac{\alpha^2+\beta^2}{\alpha\beta}xy-\frac{\beta^2+\gamma^2}{\beta\gamma}yz-\frac{\gamma^2+\alpha^2}{\gamma\alpha}zx=0,$$
where $\alpha, \beta, \gamma$ are parameters, that are the trilinear coordinates of some point $P$.
How can I tell from the equation of the conic in which cases it will be an ellipse, a parabola or a hyperbola?
From experimenting with dynamic geometry software, I think the conic should be an ellipse if $P$ lies inside the reference triangle, i.e., $\alpha, \beta, \gamma$ are all positive, and a hyperbola if $P$ lies outside the reference triangle, i.e., at least one of $\alpha, \beta, \gamma$ is negative.
Is this true? If so, how can I tell this from the equation?