In a paper I'm reading about ellipses they talk a lot about "pencils of conics", after looking around on the web to learn more like this website:
I found some simple examples and thought that maybe in the case of ellipses I understood enough to keep reading, but now I'm not so sure. From what I understand from the planetmath website is that any two conics (in my case ellipses) $H_1$ and $H_2$ generate a pencil by their four intersection points, that then generate four lines $L_1$, $L_2$, $L_3$, and $L_4$ and the pencil is $pL_1L_2=qL_3L_4$ where $p$ and $q$ are free parameters. All of this I feel comfortable with, until the paper says the following (paraphrasing a bit):
"... positions 10, 9, and 7 are separated according to the nature of the degenerate conics in the pencil $\alpha A+B$ in the following way:
position 10: if there are no degenerate conics.
position 9: if they are a pair of real lines.
position 7: if the only degenerate conic in the pencil is a double real line."
Where $\alpha$ is a scalar and $A$ and $B$ are two $3\times3$ matrix representation of two different ellipses. I don't understand how once we make a new $3\times3$ matrix $C=\alpha A+B$ that corresponds to a pencil? I though we needed two conics to make a pencil and so we would need another $3\times3$ matrix besides $C$ to generate a pencil. If I'm not being clear then please let me know I can always elaborate. Or if it's clear that I just need to read more I would be grateful for resources. Also, the paper I'm reading can be found here: