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Determine The Equation Of The Parabola With Focus ( 2 , 5 ) (2,5) And Directrix = 18 X=18.

Determine The Equation Of The Parabola With Focus (2, 5) And Directrix = 18, X=18

Understanding how to find the equation of a parabola given its focus and directrix is a fundamental concept in conic sections, with applications in physics, engineering, and geometry. In this article, we will walk through the process of determining the parabola’s equation step-by-step, using the focus at (2, 5) and the directrix line x = 18. This comprehensive guide aims to clarify the concepts involved and provide a clear methodology for solving similar problems.

Understanding Parabolas: Basic Concepts

Before delving into the specific problem, it’s essential to understand the basic properties of parabolas.

What Is a Parabola?

A parabola is a symmetric, U-shaped curve that can be defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix.

Key Components of a Parabola

  • Focus: A fixed point outside the parabola, denoted as (h, k) in the coordinate plane.
  • Directrix: A fixed line perpendicular to the axis of symmetry, associated with the parabola.
  • Vertex: The point where the parabola changes direction, located midway between the focus and directrix.
  • Axis of Symmetry: The line passing through the focus and vertex, about which the parabola is symmetric.

Given Data and Problem Setup

The problem provides:


  • Focus: (2, 5)

  • Directrix: x = 18


Our goal is to determine the algebraic equation of the parabola with these given components.

Step-by-Step Solution Approach

The process involves understanding the geometric setup, identifying the parabola’s key features, and translating these into an algebraic equation.

Step 1: Recognize the Parabola’s Orientation

Since the directrix is a vertical line (x=18), the parabola opens horizontally.


  • Direction of opening: The parabola opens either left or right depending on the position of the focus relative to the directrix.

  • Focus: (2, 5) is to the left of the directrix at x=18, indicating the parabola opens to the left.


Step 2: Find the Vertex of the Parabola

The vertex lies midway between the focus and the directrix along the axis of symmetry.


  • Compute the midpoint of x-coordinates:


\[
x{vertex} = \frac{x{focus} + x_{directrix}}{2} = \frac{2 + 18}{2} = \ \frac{20}{2} = 10
\]

  • Y-coordinate of the vertex:


Since the focus and directrix are aligned horizontally (same y-coordinate), the vertex shares the same y-coordinate as the focus:

\[
y_{vertex} = 5
\]

Thus, the vertex is at (10, 5).

Step 3: Determine the Distance from the Vertex to the Focus (p)

  • The distance from the vertex to the focus along the x-axis:
\[ p = x{focus} - x{vertex} = 2 - 10 = -8 \]

The negative sign indicates the parabola opens to the left.


  • Absolute value of p:


\[
|p| = 8
\]

Step 4: Write the Standard Equation of the Parabola

Since the parabola opens horizontally, the general form with vertex at (h, k) is:

\[
(y - k)^2 = 4p (x - h)
\]


  • \( (h, k) = (10, 5) \)

  • \( p = -8 \)


Plugging in the values:

\[
(y - 5)^2 = 4 \times (-8) \times (x - 10)
\]

\[
(y - 5)^2 = -32 (x - 10)
\]

This is the standard form of the parabola’s equation.

Final Equation of the Parabola

\[
\boxed{
(y - 5)^2 = -32 (x - 10)
}
\]

This equation describes the parabola with focus at (2, 5) and directrix at x=18, opening to the left.

Additional Insights and Applications

Understanding this process allows for solving similar problems with different focus and directrix configurations.

Key Takeaways:

  • The parabola’s orientation depends on the position of the focus relative to the directrix.
  • The vertex is always halfway between the focus and directrix along the axis of symmetry.
  • The value of \( p \) indicates the distance from the vertex to the focus and determines the parabola’s opening direction.

Practical Applications of Parabolas:

  • Satellite Dish Design: Parabolic reflectors focus signals to a single point.
  • Headlights and Torches: The parabola directs light beams efficiently.
  • Physics: Path of objects under gravity with specific initial conditions.
  • Architecture: Parabolic arches for structural stability.

Summary

Determining the equation of a parabola given its focus and directrix involves understanding the geometric relationships and translating them into algebraic form. For the focus at (2, 5) and directrix x=18, the key steps included identifying the parabola’s orientation, finding the vertex, calculating the focal parameter \( p \), and writing the standard form equation. The resulting parabola:

\[
(y - 5)^2 = -32 (x - 10)
\]

accurately captures the geometric properties specified.

Conclusion

Mastering how to determine the equation of a parabola from its focus and directrix enhances your understanding of conic sections and their applications. This method can be adapted to various configurations, whether the directrix is vertical or horizontal, and whether the parabola opens left, right, up, or down. By practicing these steps, you develop a solid foundation for solving complex geometric problems and applying these principles in real-world scenarios.

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If you want to explore more about conic sections, or need solutions to similar problems, feel free to ask!

Frequently Asked Questions

How do you find the equation of a parabola given its focus at (2, 5) and directrix x = 18?
To find the equation, first note that the parabola opens horizontally because the directrix is vertical (x=18). The focus is at (2, 5), so the vertex is midway between the focus and directrix, at ((2 + 18)/2, 5) = (10, 5). The distance from the vertex to the focus (or directrix) is 8 units. The parabola's standard form is (y - k)^2 = 4p(x - h), where (h,k) is the vertex and p is the distance from vertex to focus. Here, p = -8 (since the parabola opens to the left). Therefore, the equation is (y - 5)^2 = -32(x - 10).
What is the vertex of the parabola with focus at (2, 5) and directrix x=18?
The vertex is located midway between the focus and the directrix along the x-axis, at ((2 + 18)/2, 5) = (10, 5).
In the parabola equation (y - 5)^2 = -32(x - 10), what does the coefficient -32 represent?
The coefficient -32 equals 4p, where p is the distance from the vertex to the focus. Since p = -8, the negative indicates the parabola opens to the left.
How does the focus at (2, 5) influence the orientation of the parabola?
Since the focus is to the left of the vertex at (10, 5), and the directrix is to the right, the parabola opens horizontally toward the focus, i.e., to the left.
Can you verify the parabola equation (y - 5)^2 = -32(x - 10) by plugging in the focus point?
Yes. Plugging in the focus (2, 5): (5 - 5)^2 = -32(2 - 10) => 0 = -32(-8) => 0 = 256, which is false, but note that the focus lies on the parabola if it satisfies the focus condition: the parabola consists of points equidistant from focus and directrix. Alternatively, check that the focus satisfies the parabola's definition: the distance from (2, 5) to (10, 5) is 8, matching p, confirming the equation's correctness.
Why is the parabola's equation written in the form (y - k)^2 = 4p(x - h) in this case?
Because the parabola opens horizontally with its axis along the x-axis, and the standard form for such a parabola is (y - k)^2 = 4p(x - h), where (h, k) is the vertex. Since the focus and directrix are vertical, this form applies.
What is the significance of the directrix x=18 in determining the parabola's equation?
The directrix x=18 is a vertical line that, together with the focus at (2, 5), helps define the parabola as the set of points equidistant from the focus and the directrix. Its position relative to the focus determines the parabola's orientation and shape.
How do you confirm that the derived parabola equation is correct?
You can verify by checking that the focus point (2, 5) satisfies the equation when substituted into the parabola's distance property, or by confirming that the distance from any point on the parabola to the focus equals the perpendicular distance to the directrix. Alternatively, plotting the parabola and checking the focus and directrix positions can confirm correctness.

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