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

Verified step by step guidance

1. Identify the general form of a parabola with a horizontal axis of symmetry. The standard form is: $y^2\; =\; 4p(x\; -\; h)$, where $(h,\; k)$ is the vertex, and $p$ is the distance from the vertex to the focus or the directrix.
2. Determine the vertex of the parabola. The vertex lies midway between the focus and the directrix. Since the focus is at $(12,\; 0)$ and the directrix is $x\; =\; -12$, calculate the midpoint between these two points to find the vertex.
3. Calculate the value of $p$, which is the distance from the vertex to the focus. Use the formula $p\; =\; |x\_\{focus\}\; -\; x\_\{vertex\}|$.
4. Substitute the vertex coordinates $(h,\; k)$ and the value of $p$ into the standard form equation $y^2\; =\; 4p(x\; -\; h)$.
5. Simplify the equation to express it in standard form, ensuring all terms are properly arranged and simplified.

Key Concepts

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. This form allows for easy identification of the parabola's orientation and key features, such as the focus and directrix.

Focus and Directrix

In the context of parabolas, the focus is a fixed point from which distances to points on the parabola are measured, while the directrix is a line that is perpendicular to the axis of symmetry. The parabola is defined as the set of points equidistant from the focus and the directrix, which is crucial for determining its equation.

Vertex of the Parabola

The vertex of a parabola is the point where it changes direction and is located midway between the focus and the directrix. For a parabola that opens horizontally, the vertex can be calculated as the midpoint of the focus and the directrix, which is essential for writing the standard form of the parabola's equation.