Parabola with Vertex at (5, -2) and Focus at (5, -4): An Analysis

In this article, we will explore the detailed steps and equations of a parabola with a vertex at (5, -2) and a focus at (5, -4). We will delve into the equation of the parabola, its directrix, and the relationship between the vertex, focus, and any point on the parabola.

1. Understanding the Structure of the Parabola

We start by identifying the given information: the vertex is at (5, -2), and the focus is at (5, -4). Since the x-coordinates of the vertex and the focus are the same, the axis of symmetry for this parabola is x 5. Furthermore, since the y-coordinate of the focus is less than that of the vertex, the parabola opens downwards.

2. Deriving the Equation of the Parabola

The standard form of a vertical parabola is given by either x - h^2 4py - k or y - k^2 4px - h. Given the vertex and the focus, we can determine the parameters of the parabola.

The vertex form is x - h^2 4py - k. Here, the parameter h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex. Thus, h 5 and k -2.

The distance between the vertex and the focus, denoted as |p|, can be calculated as the absolute difference in the y-coordinates:

p |k - (-2)| |-2 - (-4)| 2

Since the parabola opens downwards, the value of p is negative: p -2.

Substituting h 5, k -2, and p -2 into the equation, we get:

x - 5^2 4(-2)y - (-2)

x - 25 -8y 2

x - 25 8y 2

x - 25 8y 2

Rearranging the terms, we obtain the final equation of the parabola:

x - 5^2 -8y - 2

3. Deriving the Directrix and Latus Rectum

The directrix of a parabola is a line that is equidistant from the vertex as the vertex is from the focus. Given that the vertex is (5, -2) and the focus is (5, -4), the directrix must be at a distance of 2 units above the vertex along the same line x 5.

Thus, the equation of the directrix is:

y 0

The length of the latus rectum, which is the chord that passes through the focus, is given by |4p|. Since p -2, we have:

|4(-2)| 8

The endpoints of the latus rectum can be found using the same formula y 0, substituting x 5 (4p/2) 5 - 4:

x 5 - 4 1

Hence, the endpoints of the latus rectum are (1, -4) and (1, 0).

4. Properties of the Parabola

Any point P (x, y) on the parabola is equidistant from the focus and the directrix. Let's verify this through the derived equation:

(x - 5)^2y - (-4)^2 y^2

(x - 5)^2y - 16 y^2

(x - 5)^2 16 y^2

Rearranging, we get:

(x - 5)^2 y^2 16

This confirms that any point on the parabola satisfies the condition of being equidistant from the focus and the directrix.