Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (7, 0); Directrix: x = - 7

Verified step by step guidance

1. Identify the orientation of the parabola based on the focus and directrix. Since the focus is at (7, 0) and the directrix is the vertical line x = -7, the parabola opens horizontally (either left or right).
2. Recall the standard form of a horizontally opening parabola: \[(y - k)^2 = 4p(x - h)\] where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus (or directrix).
3. Find the vertex \((h, k)\) by calculating the midpoint between the focus and the directrix. The x-coordinate of the vertex is the average of 7 and -7, and the y-coordinate is the same as the focus's y-coordinate.
4. Calculate the value of \(p\), which is the distance from the vertex to the focus along the x-axis. Since the parabola opens horizontally, \(p\) is positive if it opens to the right and negative if it opens to the left.
5. Substitute the vertex coordinates \((h, k)\) and the value of \(p\) into the standard form equation \[(y - k)^2 = 4p(x - h)\] to write the equation of the parabola in standard form.

Key Concepts

Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola by using the distance formula between any point on the parabola and these two elements.

Standard Form of a Parabola Equation

The standard form depends on the orientation of the parabola. For a parabola opening horizontally, the equation is (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. Identifying the vertex and p is essential to write the equation correctly.

Using the Focus and Directrix to Find the Vertex and p

The vertex lies midway between the focus and the directrix. The distance p is the distance from the vertex to the focus (positive if the parabola opens towards the focus). Calculating these values from the given focus and directrix allows you to determine the parabola's equation.