1. Fundamental Definition of a Parabola

A parabola is defined as the locus of all points $P(x, y)$ in a plane such that its distance from a fixed point, called the focus ($S$), is equal to its perpendicular distance from a fixed line, called the directrix ($L$). Mathematically, this definition is expressed as $PS = PM$, where $M$ is the foot of the perpendicular from $P$ to the directrix. Squaring both sides, we get $PS^2 = PM^2$, which helps in deriving the algebraic equation of the parabola.

2. Identifying Key Elements: Vertex, Directrix, and Focus

We are given:

• Vertex ($V$): $(5