Conics: The line $\mathbf{l} = C \mathbf{x}$ passes through $\mathbf{x}$, since $\mathbf{l}^T \mathbf{x} = \mathbf{x}^T C \mathbf{x} = 0$

Question

I'm doing some work with conics. I am told that the line $\mathbf{l}$ tangent to a conic coefficient matrix $C$ at a point $\mathbf{x}$ on $C$ is given by $\mathbf{l} = C\mathbf{x}$. It is then said that the line $\mathbf{l} = C \mathbf{x}$ passes through $\mathbf{x}$, since $\mathbf{l}^T \mathbf{x} = \mathbf{x}^T C \mathbf{x} = 0$. The justification that this is true since $\mathbf{l}^T \mathbf{x} = \mathbf{x}^T C \mathbf{x} = 0$ is not clear to me. I also tried to understand this justification using matrix operations, but in $\mathbf{l}^T \mathbf{x} = \mathbf{x}^T C \mathbf{x}$, it is not clear where the $\mathbf{x}^T$ in $\mathbf{x}^T C \mathbf{x}$ came from? Please help me understand this. I think there is a linear algebra and geometric interpretation that I am not understanding here. Thank you.