Let $E_1 E_2$ and $F_1 F_2$ be the chords of $S$ passing through the point $P_0(1,1)$ and parallel to the $x$-axis and the $y$ axis, respectively. Let $G_1 G_2$ be the chord of $S$ passing through $P_0$ and having slope -1 . Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3$, and the tangents to $S$ at $G_1$ and $G_2$ meet at $\mathrm{G}_3$. Then, the points $\mathrm{E}_3, \mathrm{~F}_3$, and $\mathrm{G}_3$ lie on the curve