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Let $C$ be a smooth projective plane curve, let $P \in C(k)$, and let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$, in the dual plane $(\mathbb{P}^2)^*$ (the plane that parameterizes lines in $\mathbb{P}^2$).Let $P^*$ denote the line in $(\mathbb{P}^2)^*$ that parameterizes the lines passing through the point $P$. Note that (by definition of $C^*$), the line $\ell$ corresponds to ta point of $C^*$.

My question is, is $P^*$ the tangent line to $C^*$ at its point $\ell$? I am able to check that the answer is yes in the case of a conic, but I do not know how to show this in general.

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From Gelfand, Kapranov & Zelevinsky: Discriminants, Resultants and multidimensional determinants.

A local parametric equation of $C$ has the form $x=x(t), y=y(t)$ where $t$ is a local coordinate on $C$ and $x(t), y(t)$ are analytic functions. By definition the dual curve $C^{\vee}$ has the parameterization $p=p(t), q=q(t)$, where $p(t)x+q(t)y+1=0$ is the affine equation of the tangent line to $C$ at $(x(t),y(t))$. Hence the parametric representation of $C^{\vee}$ has the form $p(t)=\frac{-y'(t)}{x'(t)y(t)-x(t)y'(t)}, q(t)=\frac{x'(t)}{x'(t)y(t)-x(t)y'(t)}$.

This can be applied to give biduality for plane curves. My point being, the equation for the tangent line $px+qy+1=0$ can be read both ways, as the tangent line of $C$ and of $C^{\vee}$.

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