Problems from Invariants of the quadrics and Euclidean classification

Let's consider that $4 x^{2} + 9 y^{2} + 16 z^{2} + 12 x y + 16 x z + 24 y z + 2 x + 4 y + 6 z + 1 = 0$ . Classify the quadric.

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Development:

The matrix associated with the equation of the quadric is: $\overset{―}{A} = [\begin{matrix} 4 & 6 & 8 & 1 \\ 6 & 9 & 12 & 2 \\ 8 & 12 & 16 & 3 \\ 1 & 2 & 3 & 1 \end{matrix}]$ We are going to calculate, now, its euclidean invariants. $d e t (x \cdot I - \overset{―}{A}) = x^{4} - 30 x^{3} + 15 x^{2} + 6 x$ $d e t (x \cdot I - A) = x^{3} - 29 x^{2}$ Therefore, we have : ${\begin{cases} D_{4} = 0 \\ D_{3} = - 6 \\ D_{2} = 15 \\ D_{1} = 30 \end{cases}$ ${\begin{cases} d_{3} = 0 \\ d_{2} = 0 \\ d_{1} = 29 \end{cases}$

The index of the quadric is $0$ due to the fact that the condition $d_{1} \cdot d_{3} < 0$ and $d_{2} < 0$ is not satisfied.

Solution:

As $D_{4} = 0, d_{3} = 0, d_{2} = 0, D_{3} = - 6$ , for the classification algorithm we can see that it is a parabolic cylinder.

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