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. Recall that the standard form of a parabola depends on its orientation. Since the focus and directrix are vertical lines (focus at (-5, 0) and directrix at x = 5), the parabola opens horizontally.
2. Find the vertex of the parabola, which lies exactly halfway between the focus and the directrix. Calculate the midpoint of the x-coordinates of the focus and directrix: \(x_v = \frac{-5 + 5}{2} = 0\). The y-coordinate of the vertex is the same as the focus's y-coordinate, so \(y_v = 0\). Thus, the vertex is at \((0, 0)\).
3. Determine the distance \(p\) from the vertex to the focus (or directrix). Since the focus is at \((-5, 0)\) and the vertex at \((0, 0)\), \(p = -5\) (negative because the parabola opens to the left).
4. Use the standard form of a horizontally opening parabola with vertex at \((h, k)\): \[(y - k)^2 = 4p(x - h)\]. Substitute \(h = 0\), \(k = 0\), and \(p = -5\) to get \(y^2 = 4(-5)(x - 0)\).
5. Simplify the equation to get the standard form: \[y^2 = -20x\]. This is the equation of the parabola with focus \((-5, 0)\) and directrix \(x = 5\).

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 based on the given focus and directrix.

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a parabola opening left or right, the form 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.

Finding the Vertex from Focus and Directrix

The vertex lies midway between the focus and the directrix. By calculating the midpoint of the focus and the directrix line, you can determine the vertex coordinates, which are essential for writing the parabola's equation in standard form.