The Equation of the Directrix for a Parabola with Given Focus and Vertex

In this article, we will explore the process of finding the equation of the directrix of a parabola, given the coordinates of the focus and the vertex. The specific case we will analyze is a parabola with a focus at (0, 6) and a vertex at (0, 4).

Determining the Orientation of the Parabola

Given that the focus is at (0, 6) and the vertex is at (0, 4), we observe that the y-coordinate of the focus is greater than that of the vertex. This indicates that the parabola opens upwards. The orientation of a parabola plays a crucial role in understanding its geometric properties, including the location of the directrix.

Calculating the Distance from the Vertex to the Focus

The next step is to determine the distance between the vertex and the focus. This distance is denoted by ( p ). The calculation is straightforward:

[ p text{y-coordinate of focus} - text{y-coordinate of vertex} 6 - 4 2 ]

Thus, ( p ) equals 2 units.

Locating the Directrix

For a parabola that opens upwards, the directrix is located at a distance of ( p ) units below the vertex. Therefore, the y-coordinate of the directrix is:

[ text{y-coordinate of directrix} text{y-coordinate of vertex} - p 4 - 2 2 ]

Consequently, the equation of the directrix is:

[ y 2 ]

Since the directrix is perpendicular to the axis of symmetry (the y-axis in this case) and is located at a horizontal distance from the vertex, the equation simplifies to:

[ y 2 ]

Conclusion

The equation of the directrix for the given parabola is ( y 2 ). This equation represents a horizontal line that is 2 units below the vertex.

Additional Insights

In summary, to find the equation of the directrix for a parabola, we need to follow these steps:

Determine the orientation of the parabola based on the coordinates of the focus and vertex. Calculate the distance ( p ) from the vertex to the focus. Locate the directrix by subtracting ( p ) from the y-coordinate of the vertex. Write the equation of the directrix based on the y-coordinate you obtained.

Understanding these steps not only helps in solving problems related to parabolic geometry but also enhances one's knowledge of conic sections.

Related Keywords

parabola directrix focus and vertex parabola equation