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.