We assume that $X$ is between $O,Y$ and $P$ is between $O,Z.$ $V$ is the midpoint of $\overline{XY}$ and $R=OY$ denotes the radius of $\omega.$ $K$ is the orthogonal projection of $S$ on $\ell_2.$ From $\triangle SKX \sim \triangle ZYX$ and $\triangle PXZ \sim \triangle OSZ,$ we get $\frac{SK^2}{ZY^2}=\frac{SK^2}{PZ(PZ+2R)}=\frac{SX^2}{ZX^2}=\frac{R^2}{PZ^2} \Longrightarrow SK^2=\frac{PZ+2R}{PZ} \cdot R^2 \ \ (1)$ $\frac{XK}{XY}=\frac{SX}{ZX}=\frac{R}{PZ} \Longrightarrow \frac{2KV}{XY}=\frac{2(XK+\frac{1}{2}XY)}{XY}=\frac{PZ+2R}{PZ} \ \ (2)$ Combining the expressions $(1)$ and $(2),$ we obtain $SK^2=KV \cdot \frac{2R^2}{XY}.$ This means that the locus of $S$ is a parabola wih focal axis $\ell_2,$ vertex $V$ the midpoint of $\overline{XY}$ and whose latus rectum length is given by $\frac{_{2R^2}}{^{XY}}.$