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From time to time, someone posts a question amounting to setting $$ g(x,y) = a x^2 + b x y + c y^2 + d x + e y + f $$ to zero. Then either the OP or someone answering points out the significance of the determinant of $$ H = \left( \begin{array}{ccc} 2a & b & d \\ b & 2c & e \\ d & e & 2f \end{array} \right), $$ which is double $$ 4acf+edb -ae^2-cd^2-fb^2. $$

Oh, this determinant, along with the $\pm$ sign of $b^2 - 4 a c,$ tells us whether we have degenerate, empty, nondegenerate, hyperbola, ellipse, etc.

I have been trying the trick with using Google with expressions involving exponents and putting suffix site:math.stackexchange.com SEE http://meta.math.stackexchange.com/questions/19299/find-a-specific-question-on-main

but no luck so far. Part of the trouble is my lack of internet savvy, part is the separation of the different parts of the observation among th original question, the comments, and the answer(s).

Anyway, this has been posted multiple times over the years, if anyone can find one or more examples that would be very nice.

In case it helps, this has to do with the extra material I am putting at the end of my answer What's so special about the form $ax^2+2bxy+cy^2$?

with an application one day later at $x^2+y^2+9=3(x+y)+xy$ Find all pairs of real $x,y$ that meet this equation

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