This problem requires us to find the equation of a parabola given its vertex and directrix, and then use that equation to determine the sum of specific coefficients. The core concept for solving this is the fundamental definition of a parabola.

Key Concept: Definition of a Parabola

A parabola is defined as the locus of a point $P(x,y)$ that moves in a plane such that its distance from a fixed point (the focus, $S$) is equal to its perpendicular distance from a fixed line (the directrix, $L$). Mathematically, if $P(x, y)$ is a point on the parabola, $S$ is the focus, and $L$ is the directrix, then $PS = PD$, where $PD