Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence

$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\rightarrow 0$$

where $\mathcal{I}_p$ is the ideal sheaf of a point $p\in\mathbb{P}^2$. Then $c_1(\mathcal{E})=-1$ and $c_2(\mathcal{E})=1$.

Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^2$, and $-K_X$ the anti-canonical divisor of $X$. Finally, let $C\subset X$ be an irreducible curve such that $\pi(C) = L_p$ is a line through $p\in\mathbb{P}^2$.

How can we compute the intersection number $-K_X\cdot C$ ?