An Isosceles Slope

In the coordinate plane, a parabola $C$ has vertex at the origin, is symmetric about the $x$-axis, and passes through $P(1, 2)$.

1. Find the equation of the parabola $C$. 2. Points $A, B$ lie on $C$; the lines $PA, PB$ meet the $y$-axis at $M, N$ respectively, and $|PM| = |PN|$. Find the slope of the line $AB$.