Determining the Parabola Equation for Focus F711 with Directrix y4

When dealing with parabolas, understanding the relationship between the directrix and focus is crucial for determining the equation of the parabola. In this article, we will delve into the specific case where the directrix is given as y4 and the focus is at a specific point.

Understanding the Geometry of the Parabola

A parabola is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). In the provided example, the directrix is parallel to the x-axis and located below the focus. This configuration indicates that the parabola opens upward.

Identifying Key Components of the Parabola

The key components identified in the problem are the vertex (V), the focus (F), and the directrix (D). Given the directrix y 4, we can infer the orientation and placement of the parabola. Specifically:

The vertex (V) is located at x7 and y 411/2 7.5. This is the midpoint between the focus (F) and the directrix (D), forming the vertical axis of symmetry for the parabola.

The distance from the vertex to the focus (p) is calculated as 11 - 7.5 3.5. This distance is also the distance from the vertex to the directrix.

Formulating the Equation of the Parabola

The standard form of the equation for a parabola that opens upwards is given by:

(x - h)2 4p(y - k)

Where:

(h, k) are the coordinates of the vertex.

p is the distance from the vertex to the focus (or the directrix).

Given the values from our problem:

h 7

k 7.5

p 3.5

The equation of the parabola can be formulated as:

(x - 7)2 4(3.5)(y - 7.5)

After simplifying, we get:

(x - 7)2 14(y - 7.5)

This is the standard form of the parabola equation with the given parameters.

Additional Insights

Understanding the relationship between the focus (F) and the directrix (D) can help solve a variety of parabolic problems. For instance, in the provided problem, we can see that:

Directrix: y 4

Focus: Above the directrix, so the parabola opens upwards.

Distance from the vertex to the focus (p): 0

This setup, represented in the form (x - h)2 4py - k, ensures that the parabola opens correctly. Plugging in the values, we get:

h 7

0 1

y k - p 4

This implies that k 7.5 and p 3.5

These values verify the earlier formulation of the parabola equation.

Conclusion

By understanding the relationship between the focus and the directrix, we can accurately determine the equation of a parabola. The example provided illustrates how to use the given information to form the equation of a parabola in standard form. This knowledge can be applied to various real-world scenarios, from optics to engineering.

The key components and steps involved in determining the parabolic equation make the problem-solving process straightforward and accessible. Whether you're in high school math or delving into more advanced applications, a solid grasp of parabolic equations is invaluable.