Conics consisting of two points/lines makes them rank 2 While studying conics, I came across this concept and example:

Degenerate conics. If the matrix $C$ is not of full rank, then the conic is termed degenerate. Degenerate point conics include two lines (rank 2), and a repeated line (rank 1).
Example. The conic
$$C = \mathbf{l}\mathbf{m}^T + \mathbf{m} \mathbf{l}^T$$
is composed of two lines $\mathbf{l}$ and $\mathbf{m}$. Points on $\mathbf{l}$ satisfy $\mathbf{l}^T \mathbf{x} = 0$, and are on the conic since $\mathbf{x}^T C \mathbf{x} = (\mathbf{x}^T \mathbf{l})(\mathbf{m}^T \mathbf{x}) + (\mathbf{x}^T \mathbf{m})(\mathbf{l}^T \mathbf{x}) = 0$. Similarly, points satisfying $\mathbf{m}^T \mathbf{x} = 0$ also satisfy $\mathbf{x}^T C \mathbf{x} = 0$. The matrix $C$ is symmetric and has rank 2. The null vector is $\mathbf{x} = \mathbf{l} \times \mathbf{m}$ which is the intersection point of $\mathbf{l}$ and $\mathbf{m}$.
Degenerate line conics include two points (rank 2), and a repeated point (rank 1). For example, the line conic $C^* = \mathbf{x} \mathbf{y}^T + \mathbf{y} \mathbf{x}^T$ has rank 2 and consists of lines passing through either of the two points $\mathbf{x}$ and $\mathbf{y}$. Note that for matrices that are not invertible $(C^*)^* \not= C$.

I'm wondering why these conics consisting of two points/lines makes them rank 2 (and why is the repeated point for the latter rank 1)? I'd really appreciate clarification of this example. Thank you.

I was also really confused by the lack of explanation of some topics in this book. To have a better understanding of degenerate conics in 2D projective space, I had to look for alternative sources of info. Watching a visualization of conics and degenerate conics on youtube helped me out a lot. I finally stumbled upon this pdf:
https://link.springer.com/content/pdf/bbm%3A978-0-85729-046-5%2F1.pdf
It doesn't explains much more than the book, but theres a key phrase in page 3 that gave me a little bit of intuition:
"In degenerate cases, it may degenerate to two lines when $\operatorname{rank}(C) = 2$, or one repeated line when $\operatorname{rank}(C) = 1$."
Meaning that the rank is for the matrix $C$. Then I found this presentation of the University of Berkeley about Multiple view geometry:
https://inst.eecs.berkeley.edu/~ee290t/fa11/lectures/class02-geometry-and-camera-model.pptx
On slide 15 you can see a better explanation of this subject.
