Determine the Equation of a Horizontal Parabola and Calculate its Latus Rectum

When dealing with parabolas, one of the most common tasks is to determine their equations given specific points and conditions. In this article, we will explore how to find the equation of a horizontal parabola with a given vertex and a point it passes through, and then calculate the length of its latus rectum.

Given Information and Standard Form of a Parabola

Consider a parabola with a vertex at (6, 0) and passing through the point (2, 1). The axis of symmetry of the parabola is parallel to the x-axis. The standard form of the equation of a horizontally oriented parabola is given by:

y - k2 4p(x - h)

where (h, k) is the vertex and p is the distance from the vertex to the focus or the directrix.

Step 1: Substituting the Vertex

Given the vertex (6, 0), we can substitute h 6 and k 0 into the standard form:

y2 4p(x - 6)

Step 2: Finding the Value of p

We need to determine the value of p using the point (2, 1). Plugging this point into the equation:

12 4p(2 - 6)

This simplifies to:

1 4p(-4)

Which further simplifies to:

1 -16p

Solving for p:

p -1/16

Step 3: Final Equation of the Parabola

Now we can substitute p back into the equation:

y2 4(-1/16)(x - 6)

This simplifies to:

y2 -1/4(x - 6)

Step 4: Length of the Latus Rectum

The length of the latus rectum of a parabola is given by |4p|. Since we found that p -1/16:

Length of the latus rectum |4p| |-4/16| 1/4

Summary

The equation of the parabola is:

y2 -1/4(x - 6)

The length of the latus rectum is:

1/4

Further Insights

Understanding the properties of parabolas is crucial in many fields, including physics, engineering, and mathematics. The axis of symmetry parallel to the x-axis implies the parabola opens horizontally. The vertex (6, 0) is the point of minimum or maximum of the parabola, depending on its orientation.

For more detailed information and further exploration, additional resources such as textbooks, online tutorials, and research papers on parabolic equations can be very helpful.