Understanding Conics in Pencil

Question

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:

http://planetmath.org/pencilofconics

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:

http://www.sciencedirect.com/science/article/pii/S0167839606000033

Answer 1

OK, I looked at the paper. I think the passage you are asking about is very confusingly written. What they appear to be saying is that for this specific value, $\alpha A +B$ is the (unique) degenerate conic in the pencil. So the plural "conics" in the first line and the pronoun "they" later on are incorrect. So to sum up: 1) the pencil is $\lambda A +B$ where $\lambda$ is a variable. 2) $\alpha A +B$ is a specific conic in the pencil, the unique degenerate one. 3) the referee did a bad job.

Answer 2

$C$ is made from the two matrices $A$ and $B$ each defining a conic. Let $\alpha=0$ and you get the one, let $\alpha=\infty$ and you get the other. The whole pencil is described by varying $\alpha$. (If you don't feel comfortable with one conic appearing only in the limit $\alpha\to\infty$, consider $\lambda A+(1-\lambda) B$ instead.)

Assuming the conics meet in four points, $Ap_k=Bp_k=0$ so that $(\alpha+B)p_k=0$ and all the conics in the pencil have these points in common. In particular, there are conics in the pencil that are degenerate, i.e. pairs of lines through the four points.