Exploring the Vertex, Focus, Directrix, and Latus Rectum of a Parabola

In the context of geometry and algebra, the properties of a parabola such as its vertex, focus, directrix, and latus rectum are fundamental. This article will delve into these properties, particularly for the parabola represented by the equation x - 7^2 y - 1/2.

1. Understanding the Parabola Equation

The given equation is x - 7^2 y - 1/2. By rearranging it, we can express it in vertex form:

y (1/4)(x - 7)^2 1/2

This is a typical vertex form equation for an upward-opening parabola, where the vertex is located at (7, 1/2).

2. Vertex

The vertex of the parabola is the point where the parabola changes from increasing to decreasing or vice versa. For the equation above, the vertex is at:

(7, 1/2)

3. Focal Length and Focus

The focal length (p) is the distance from the vertex to the focus. For the given parabola, we can determine the focal length using the equation p 1/4a, where a 1/4 from the equation above. Thus:

p 1/4 * (1/4) 1/16

The focus of the parabola is located at (h, k p), where (h, k) is the vertex. Therefore, the focus is:

(7, 1/2 1/16) (7, 9/16)

4. Directrix

The directrix is a line that is perpendicular to the axis of symmetry and is at a distance of p from the vertex on the opposite side of the focus. The equation for the directrix of a parabola in vertex form is given by:

y k - p

For the given parabola, the directrix is:

y 1/2 - 1/16 7/16

5. Latus Rectum

The latus rectum is the line segment that passes through the focus, parallel to the directrix, and has its endpoints on the parabola. For a parabola in vertex form, the length of the latus rectum is given by 4p. Since p 1/16, the length of the latus rectum is:

4p 4 * 1/16 1/4

To find the endpoints of the latus rectum, we use the x-coordinates of the points where the latus rectum intersects the parabola. We know that the y-coordinate of the focus is 9/16. We substitute this into the parabola equation:

(1/4)(x - 7)^2 1/2 9/16

(x - 7)^2 9/16 - 1/2 9/16 - 8/16 1/16

x - 7 ±1/4

x 7 ± 1/4

Therefore, the endpoints of the latus rectum are:

(7 1/4, 9/16) (7.25, 0.5625)

(7 - 1/4, 9/16) (6.75, 0.5625)

Conclusion

Understanding the properties of a parabola, such as its vertex, focus, directrix, and latus rectum, is crucial in both geometry and algebra. For the parabola defined by the equation x - 7^2 y - 1/2, the vertex is at (7, 1/2), the focus is at (7, 9/16), the directrix is at y 7/16, and the latus rectum is a line segment of length 1/4 with endpoints at (6.75, 0.5625) and (7.25, 0.5625).