Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (12,0); Directrix: x=-12

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the intersection of a cone with a plane parallel to its side. It can be defined as the set of all points equidistant from a fixed point called the focus and a fixed line known as the directrix. The orientation of the parabola (opening left, right, up, or down) depends on the position of the focus relative to the directrix.

Standard Form of a Parabola

The standard form of a parabola that opens horizontally is given by the equation (y - k)² = 4p(x - h), where (h, k) is the vertex, and p is the distance from the vertex to the focus or directrix. For a parabola opening to the right, p is positive, while for one opening to the left, p is negative. This form allows for easy identification of the vertex and the direction in which the parabola opens.

Focus and Directrix Relationship

The focus and directrix of a parabola are crucial in determining its equation. The focus is a point from which distances to points on the parabola are measured, while the directrix is a line that serves as a reference. The distance from any point on the parabola to the focus is equal to the perpendicular distance from that point to the directrix, which is fundamental in deriving the parabola's equation.