This solution will guide you through finding the squared distance between the intersection points of two parabolas. We will start by recalling the fundamental definition of a parabola and then systematically derive the equations, find their intersection points, and finally calculate the required distance.

1. Key Concept: Definition of a Parabola

A parabola is defined as the locus of a point that moves in a plane such that its distance from a fixed point (called the focus) is equal to its perpendicular distance from a fixed line (called the directrix).

Mathematically, if $P(x,y)$ is a point on the parabola, $F(h,k)$ is the focus, and $ax+by+c=0$ is the directrix, then the equation of the parabola