I just did a question asking to classify the kind of curve of a given quadratic polynomial: $$0=3x^2+8xy+6y^2+12x+20y+17$$
I completed the square a few times and eventually (correctly) observed that it is an ellipse.
However the solution I've been given simply uses a fact that we worked out in an earlier part of the question, specifically that the signature of the quadratic part of the polynomial is 2. According to the solution this directly implies that we have an ellipse.
This is something I haven't been taught and was wondering if we are always able to use the signature to classify a curve? Can we do something similar with surfaces?
If so, then what signatures determine what kind of curve/surface?
Thanks for any help!
Too long for a comment.
First, you perhaps completed the square, but first should have gotten rid of that $\;xy\;$ term there by means of a transformation.
Second, in this precise case that you say is an ellipse, the signature doesn't help, but the sign of the determinants of the expanded and the reduced matrices, say $\;A\,,\,\,A_1\;$ resp., together with their range determine this:
$$A=\begin{pmatrix}17&6&10\\6&3&4\\10&4&6\end{pmatrix}\;\;,\;\;\;A_1=\begin{pmatrix}3&4\\4&6\end{pmatrix}$$
so
$$\begin{cases}\det A<0\\{}\\\det A_1>0\\{}\\\text{rank}\,A=3\\{}\\\text{rank}\,A_1=2\end{cases}\;\;\;\implies\;\;\text{the quadratic is an ellipse.}$$
As you can see, the signature has no business in this case. It odes have though when we have two parallel lines or the empty set.