In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Focus: (- 5, 0); Directrix: x = 5

Verified step by step guidance

1. Identify the given elements: the focus is \((-5, 0)\) and the directrix is \(x = 5\).
2. Recognize that the parabola opens horizontally because the directrix is a vertical line.
3. Use the formula for a parabola with a horizontal axis of symmetry: \((y - k)^2 = 4p(x - h)\), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.
4. Calculate the vertex \((h, k)\) as the midpoint between the focus and the directrix. The vertex is \((-5 + 5)/2, 0) = (0, 0)\).
5. Determine \(p\) as the distance from the vertex to the focus: \(p = -5 - 0 = -5\). Substitute \(h = 0\), \(k = 0\), and \(p = -5\) into the equation \((y - 0)^2 = 4(-5)(x - 0)\) to get the standard form.

Key Concepts

Parabola Definition

A parabola is a symmetric curve formed by the set of all points that are equidistant from a fixed point called the focus and a fixed line known as the directrix. The orientation of the parabola depends on the position of the focus relative to the directrix, which can be vertical or horizontal.

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. This form allows for easy identification of the vertex and the direction in which the parabola opens.

Finding the Vertex

The vertex of a parabola is the midpoint between the focus and the directrix. In this case, with the focus at (-5, 0) and the directrix at x = 5, the vertex can be calculated as the average of the x-coordinates of the focus and the directrix, resulting in the vertex being at (-5, 0). This is crucial for writing the equation in standard form.