Question
Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb P^2(k)\setminus C$. Then there's $Q\in C$, such that the tangent cone of $C$ at $Q$ passes through $P$, i.e., there's a tangent line of $C$ passing through $P$.
Background
I'm reading Motzkin & Taussky's paper pairs of matrices with property L. It seems that some stronger version of the preceding statement is involved in the proof of Theorem 3. I need to know how to show it.