Problems from Euclidean invariants of the conics

Classify the following conic: $2 x^{2} + 4 x y + y^{2} + 2 x + 4 = 0$ .

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Development:

The associated matrix is $\overset{―}{A} = [\begin{matrix} 2 & 2 & 1 \\ 2 & 1 & 0 \\ 1 & 0 & 4 \end{matrix}]$ The associated euclidean invariants are: $D_{3} = d e t \overset{―}{A} = 8 - 1 - 16 = - 9$ $d_{2} = 2 - 4 = - 2$ $d_{1} = 2 + 1 = 3$ We do not compute $D_{2}$ because the determinant of the associated matrix is other than zero.

From the classification scheme, as $D_{3} \neq 0$ and $d_{2} < 0$ , the conic is a hyperbola.

Solution:

The conic is a hyperbola.

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