Map from a curve to the projective plane

Let $(\mathbb{P}^2)^*$ denote the projective space of lines in $\mathbb{P}^2$. Let $[y_0:y_1:y_2]$ be the coordinates of $(\mathbb{P}^2)^*$, where $[y_0:y_1:y_2]$ denotes the line $$ y_0x_0+y_1x_1+y_2x_2=0. $$

If $p=[x_0:x_1:x_2]\in C$ then the line spanned by $p$ and $[1:0:0]$ is $$ 0*x_0+(x_2)*x_1+(-x_1)*x_2=0, $$ i.e., the map $\varphi$ you define sends $p$ to $[0:-x_2:x_1]$. (Note that $x_1=x_2=0$ is impossible since otherwise $p=[1:0:0]$ which is not on the curve). The image of this map is the variety $$ Z = \{l\in (\mathbb{P}^2)^*\mid [1:0:0]\in l\}, $$ i.e., the variety of all lines that passes through a fixed point (which is itself a line in $(\mathbb{P}^2)^*$). You may see it in the affine patch $\mathbb{C}^2\subset\mathbb{P}^2$ of points with $x_0\neq 0$. Then $C$ is a hyperbola, $[1:0:0]$ is the origin and the corresponding affine patch of the variety $Z$ is the set of all lines in $\mathbb{C}^2$ that passes through the origin.

The induced map between the rings is $k[y_0,y_1,y_2]\rightarrow k[x_0,x_1,x_2]/(x_0^2+x_1^2-x_2^2)$ which maps $y_0\mapsto 0, y_1\mapsto x_2, y_2\mapsto -x_1$.