# a technique of parameterization fixing a point P a taking lines through P

Maybe someone knows this technique. Given $F\left( {x,y} \right) \in {\Bbb C}\left[ {x,y} \right]$, let $C = \left\{ {\left( {x,y} \right) \in {\Bbb C};F\left( {x,y} \right) = 0} \right\}$. Take a point $P \in C$ , consider the set L of all the lines that pass through $P$, clearly those lines are determined by their slope $\lambda$. Let now any $l \in L$ be any line through P. In some cases we can parameterize. Assume that for some reason there is a point of intersection of $l$ with $C$ , let´s call it $c(l)$ such that $c(l) = c(\lambda)= (c_x(\lambda),c_y(\lambda))$ Then clearly we have the parametrization, since $c(\lambda) \in C$ so $F(c(\lambda)) = 0$

I have two questions about this, how can I prove that if there exist some point P, such that for any line that pass through P , there exist only one point (other than the fixed P), then I can write that intersection in terms of rational functions of the parameter $\lambda$.

And also how can I prove using this technique the following problem? Let $F(x,y) \in$ ${\Bbb C}\left[ {x,y} \right]$ of degree 3 , such that $F , F_x , F_y$ have a common zero , then there exist non constant polynomials $f\left( t \right),g\left( t \right) \in {\Bbb C}\left( t \right)$ such that $F(f(t),g(t))=0$

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