Given a curve (smooth, projective, irreducible) $X$ in $\mathbb{CP}^2$, this curve meets all other curves in the same space. Generically, it will meet a line (a copy of $\mathbb{CP}^1$ in $\mathbb{CP}^2$) in deg$X$ points. My question is: is there a line which meets $X$ in exactly one point? A separate question which would imply an affirmative answer to the first, does $X$ have a tangent line that is not a secant line?
EDIT: Actually, by Bezout's theorem, I think the two questions are the same.
For a degree 2 curve any tangent line will intersect the curve at exactly one point. For a generic degree 3 curve (in other words an elliptic curve), most tangent lines will intersect the curve at a third point, but there will be a single point for which the tangent line intersects it there with multiplicity 3. This point is the identity point when we put the group structure on an elliptic curve. For higher degree curves the answer is no in general.