The equation of a conic in ordinary
point coordinates is a quadratic:
axx + bxy + cyy + dx + ey + f = 0.
By appropriate adjustment of the x- and
y- coordinates, we can transform this to
axx + bxy + cyy + f = 0.
Similarly, the homogenous equation
axx + bxy + cyy + dxz + eyz + fzz = 0
can be transformed into
axx + bxy + cyy + fzz = 0.
We can differentiate this and find that
the tangent lines have equations of form
ux + vy + wz = 0,
After applying some algebra to u, v and w,
we find that
bb-4ac
cuu - buv + avv - ------ww = 0.
4f
So the equation of a conic in line
coordinates is also a quadratic (and we
could prove on this basis the identity
of point and line conics).
Thus we have a transformation (a,b,c,f) -> (c,-b,a,(4ac-bb)/4f) which converts the coefficients of the point-coordinate representation of a conic to the coefficients of the line-coordinate representation of a conic.
By the principle of duality, this transformation will also work to convert the equation of a conic from line coordinates to point coordinates.
In other words, the conic
auu + buv + cvv + fww = 0
can be written in point coordinates as
bb-4ac
cxx - bxy + ayy - ------ zz = 0.
4f
We know that transforming an equation from point coordinates to line coordinates and back again does not affect the equation. Similarly, the above transformation is its own inverse, as we can easily verify.
Because the quantity bb-4ac is invariant under this transformation, we can use it to distinguish between ellipse, parabola and hyperbola in line coordinates just as in point coordinates.