Division Points of a Square Lying on a Parabola

As shown in the figure, $OABC$ is a square in the coordinate plane, where $O$ is the origin, $A = (10,0)$, and $C = (0,10)$. The segments $OA$ and $AB$ are each divided into ten equal parts, with division points $A_1, A_2, \dots, A_9$ and $B_1, B_2, \dots, B_9$ respectively. For each integer $i$ with $1 \leqslant i \leqslant 9$, the segment $OB_i$ is drawn, and the line through $A_i$ perpendicular to the $x$-axis meets $OB_i$ at the point $P_i$.

1. Prove that the points $P_i$ ($1 \leqslant i \leqslant 9$) all lie on one parabola, and find the equation of this parabola $E$. 2. A line $l$ through the point $C$ intersects the parabola $E$ at two distinct points $M$ and $N$. If the ratio of the areas of $\triangle OCM$ and $\triangle OCN$ is $4:1$, find the equation of the line $l$.