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**Here is what I know:**

1) All quadratic curves of 2 variables correspond to a conic section.

[tex]ax^2 + 2bxy +cy^2 + 2dx + 2fy + g = 0[/tex]

[tex] a, b, c[/tex] are not all [tex]0[/tex]

2) The definitions of parabola (in terms of a directrix and focus), ellipse (in terms of 2 foci), hyperbola (in terms of directrix and focus).

3) The determinate of a 2x2 matrix is the area of the parallelogram formed by the 2 row vectors.

**Question:**

The above quadratic equation can be found to be either an ellipse, parabola or hyperbola depending on the value of the determinate

[tex]\left| \begin{array}{ccc}

\ a & b \\

b & c\end{array} \right|[/tex]

I haven't seen any sort of derivation, or even a hint, as how to arrive at the significance of this determinate.

Can anyone point me to one?

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