Finding The Equation Of A Parabola Focus (-3,0) Directrix X=3

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The fascinating world of conic sections introduces us to the parabola, a U-shaped curve defined by a unique geometric property. Understanding parabolas is crucial in various fields, from optics and antenna design to projectile motion analysis. This article delves into the specifics of finding the equation of a parabola given its focus and directrix. We'll focus on the scenario where the focus is at (-3, 0) and the directrix is the line x = 3. Let's embark on this mathematical journey to unravel the equation that governs this particular parabola.

Understanding the Parabola: Focus, Directrix, and Definition

Before diving into the equation, let's solidify our understanding of the key components of a parabola. A parabola is defined as the set of all points that are equidistant to a fixed point, called the focus, and a fixed line, called the directrix. The focus lies inside the curve of the parabola, while the directrix lies outside. The line passing through the focus and perpendicular to the directrix is the axis of symmetry of the parabola. The point where the parabola intersects its axis of symmetry is called the vertex, which is the midpoint between the focus and the directrix.

In our case, the focus is given as (-3, 0), and the directrix is the vertical line x = 3. This tells us that the parabola will open to the left because the focus is to the left of the directrix. The axis of symmetry is the x-axis (y = 0), and the vertex is located at the midpoint between the focus and the directrix. To find the vertex, we can average the x-coordinates of the focus and a point on the directrix (for example, (3, 0)). The x-coordinate of the vertex is ((-3) + 3) / 2 = 0. The y-coordinate of the vertex is the same as the y-coordinate of the focus and the point on the directrix, which is 0. Therefore, the vertex of this parabola is at (0, 0), the origin.

Knowing the focus, directrix, and vertex allows us to visualize the parabola's orientation and position in the coordinate plane. This understanding is crucial for determining the correct form of the equation.

Deriving the Equation of the Parabola

Now that we have a clear picture of the parabola's geometry, we can derive its equation using the fundamental definition: the distance from any point (x, y) on the parabola to the focus is equal to the distance from that point to the directrix. Let's break down this process step-by-step.

  1. Distance to the Focus: The distance between a point (x, y) on the parabola and the focus (-3, 0) can be calculated using the distance formula:

    √((x - (-3))^2 + (y - 0)^2) = √((x + 3)^2 + y^2)

  2. Distance to the Directrix: The distance between a point (x, y) and the directrix x = 3 is the perpendicular distance, which is simply the absolute difference in their x-coordinates:

    |x - 3|

  3. Equating the Distances: According to the definition of a parabola, these two distances must be equal:

    √((x + 3)^2 + y^2) = |x - 3|

  4. Squaring Both Sides: To eliminate the square root, we square both sides of the equation:

    (x + 3)^2 + y^2 = (x - 3)^2

  5. Expanding the Squares: Expand the squared terms:

    x^2 + 6x + 9 + y^2 = x^2 - 6x + 9

  6. Simplifying the Equation: Notice that x^2 and 9 appear on both sides of the equation, so we can cancel them out. This leaves us with:

    6x + y^2 = -6x

  7. Isolating the y^2 Term: To get the equation in the standard form, we isolate the y^2 term by subtracting 6x from both sides:

    y^2 = -12x

Therefore, the equation of the parabola with a focus at (-3, 0) and directrix x = 3 is y^2 = -12x. This equation represents a parabola opening to the left, with its vertex at the origin and an axis of symmetry along the x-axis.

Matching the Equation to the Given Options

Now that we have derived the equation y^2 = -12x, let's compare it to the options provided:

A. x^2 = -12y B. x^2 = 3y C. y^2 = 3x D. y^2 = -12x

Clearly, option D, y^2 = -12x, matches the equation we derived. Therefore, this is the correct equation for the parabola.

Key Takeaways and the Standard Form

This exercise highlights the importance of understanding the fundamental definition of a parabola and how it relates to its equation. The standard form of a parabola's equation depends on its orientation:

  • Parabola opening to the right: y^2 = 4px (focus at (p, 0), directrix x = -p)
  • Parabola opening to the left: y^2 = -4px (focus at (-p, 0), directrix x = p)
  • Parabola opening upwards: x^2 = 4py (focus at (0, p), directrix y = -p)
  • Parabola opening downwards: x^2 = -4py (focus at (0, -p), directrix y = p)

In our case, the equation y^2 = -12x fits the standard form y^2 = -4px, where -4p = -12, which means p = 3. This confirms that the focus is at (-3, 0) and the directrix is x = 3, as given in the problem.

Understanding these standard forms allows you to quickly identify the orientation, vertex, focus, and directrix of a parabola given its equation. Conversely, knowing the focus and directrix enables you to readily determine the equation.

Applications and Further Exploration

Parabolas have numerous applications in real-world scenarios. Their reflective property, where parallel rays of light or radio waves entering a parabola are reflected to the focus, is used in satellite dishes, telescopes, and car headlights. The path of a projectile, neglecting air resistance, is also parabolic.

Further exploration of parabolas can involve:

  • Finding the equation of a parabola given different information: For example, given the vertex and focus, or the vertex and directrix.
  • Analyzing parabolas with vertices not at the origin: This involves transformations of the standard equations.
  • Exploring the relationship between parabolas and other conic sections: Such as ellipses and hyperbolas.
  • Investigating the applications of parabolas in various fields: Like physics, engineering, and architecture.

Conclusion

In this article, we successfully determined the equation of the parabola with a focus at (-3, 0) and a directrix of x = 3. By applying the fundamental definition of a parabola and using algebraic manipulation, we arrived at the equation y^2 = -12x. This exercise not only reinforces our understanding of parabolas but also highlights the power of mathematical reasoning in solving geometric problems. By grasping the core concepts and practicing problem-solving, you can confidently navigate the world of conic sections and their diverse applications.