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Question. Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or have tangency of order $k$ (for any $k$ between 3 and $d$) with $C$? Probably this can be done for small $d$ using the equation for $C$, but I would like to find out if there is a formula for general $d$. Thanks

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# Pencil of lines and degree $d$ curve in $\mathbb{CP}^2$

Let $C$ be a generic smooth curve of degree $d$ in $\mathbb{CP}^2$, and let $P$ be an arbitrary point away from this curve. How many lines are there through point $P$ that are tangent, or have tangency of order $k$ (for any $k$ between 3 and $d$) with $C$? Probably this can be done for small $d$ using the equation for $C$, but I would like to find out if there is a formula for general $d$. Thanks