Union of lines in $\mathbb{P}^3$ is closed

Question

Let $H=\mathbb{P}^2$, $X=V(y^2z-x^3)\subset H$ and $p\in \mathbb{P}^3\setminus H$. Let $C$ be the union of all lines passing through $p$ and a point of $X$. I would like to find the equations defining $C$ to show that it is closed. How may I do this?