Let be $L$ tangent on P to a nonsingular cubic curve $C$. Show that L intersects C at a unique single point.

Question

I have to do it using Bezout's theorem. I have a question in my solution, Can I asumme that $I(P,L \bigcap C)=2$ because $L$ is tangent on $P$ to C?... If it is like this, the Bezout's theorem would implies that there is only another point $P_1$ with $I(P_1,L\bigcap C)=1$ ....If it is not like this... can someone help me?

.. thanks...