In Exercises 17–30, 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. Step 1: Understand the definition of a parabola. A parabola is the set of all points that are equidistant from a fixed point called the focus and a line called the directrix.
2. Step 2: Identify the given elements. The focus is (7, 0) and the directrix is x = -7. This indicates that the parabola opens horizontally.
3. Step 3: Determine the vertex of the parabola. The vertex is the midpoint between the focus and the directrix. Calculate the midpoint of the x-coordinates of the focus and directrix: (7 + (-7))/2 = 0. Thus, the vertex is at (0, 0).
4. Step 4: Use the standard form of a parabola that opens horizontally. The standard form is (y - k)^2 = 4p(x - h), where (h, k) is the vertex. Here, (h, k) = (0, 0).
5. Step 5: Calculate the value of p. The distance from the vertex to the focus is p. Since the focus is at (7, 0) and the vertex is at (0, 0), p = 7. Substitute p into the equation to get the standard form.

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

The focus of a parabola 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 of the parabola. 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 a 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 with a focus at (7, 0) and a directrix at x = -7, the vertex can be found by averaging the x-coordinates of the focus and the directrix, which is essential for writing the equation in standard form.