Finding The Equation Of A Parabola With Focus (0, 3) And Directrix Y = -3

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In the realm of conic sections, the parabola stands out as a fundamental shape with numerous applications in physics, engineering, and mathematics. Understanding how to determine the equation of a parabola given its focus and directrix is a crucial skill. In this article, we will delve into the process of finding the equation of a parabola, focusing on a specific example: a parabola with a focus at (0, 3) and a directrix of y = -3. We'll break down the concept of a parabola, its key properties, and then walk through the steps to derive its equation. By the end of this guide, you'll have a solid grasp of how to tackle similar problems and a deeper appreciation for the geometry of parabolas.

Understanding the Parabola

Before we jump into the specific problem, let's establish a clear understanding of what a parabola is. A parabola is defined as the set of all points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix. This definition is the cornerstone of our approach to finding the equation.

  • Focus: The focus is a point inside the curve of the parabola. It plays a crucial role in determining the shape and orientation of the parabola.
  • Directrix: The directrix is a line outside the curve of the parabola. It is equally important as the focus in defining the parabola's shape.
  • Vertex: The vertex is the point on the parabola that is closest to both the focus and the directrix. It lies exactly midway between them.
  • Axis of Symmetry: The axis of symmetry is the line that passes through the focus and the vertex, and it is perpendicular to the directrix. The parabola is symmetric about this line.

In our case, the focus is given as (0, 3) and the directrix is the line y = -3. Visualizing these elements on a coordinate plane is a helpful first step. We can immediately see that the vertex will lie midway between the focus and the directrix, which will be at the point (0, 0). The axis of symmetry will be the y-axis, since it passes through the focus (0, 3) and is perpendicular to the directrix y = -3.

Deriving the Equation of the Parabola

The definition of a parabola provides us with the key to finding its equation. Let (x, y) be any point on the parabola. By definition, the distance from (x, y) to the focus (0, 3) must be equal to the distance from (x, y) to the directrix y = -3. We can express these distances using the distance formula and the formula for the distance between a point and a line.

1. Distance to the Focus

The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:

distance = √((x2 - x1)² + (y2 - y1)²)

In our case, the distance between the point (x, y) on the parabola and the focus (0, 3) is:

distance_to_focus = √((x - 0)² + (y - 3)²)
                 = √(x² + (y - 3)²)

2. Distance to the Directrix

The distance between a point (x, y) and a line y = c is given by the absolute value of the difference in their y-coordinates:

distance = |y - c|

In our case, the distance between the point (x, y) on the parabola and the directrix y = -3 is:

distance_to_directrix = |y - (-3)|
                      = |y + 3|

3. Equating the Distances

Now, we apply the definition of the parabola: the distance to the focus equals the distance to the directrix. This gives us the equation:

√(x² + (y - 3)²) = |y + 3|

To eliminate the square root and the absolute value, we square both sides of the equation:

(√(x² + (y - 3)²))² = (|y + 3|)²
x² + (y - 3)² = (y + 3)²

4. Simplifying the Equation

Next, we expand the squared terms:

x² + (y² - 6y + 9) = (y² + 6y + 9)

Now, we simplify by canceling out the y² and 9 terms on both sides:

x² - 6y = 6y

Add 6y to both sides:

x² = 12y

Finally, solve for y to get the equation of the parabola in the standard form:

y = (1/12)x²

Therefore, the equation of the parabola with a focus at (0, 3) and directrix y = -3 is y = (1/12)x².

Analyzing the Result

The equation we derived, y = (1/12)x², is a standard form equation of a parabola that opens upwards. This makes sense given the position of the focus and the directrix. The focus is above the directrix, indicating that the parabola opens upwards. The coefficient (1/12) determines the width of the parabola; a smaller coefficient results in a wider parabola, while a larger coefficient results in a narrower parabola.

This equation also confirms our earlier observation that the vertex is at (0, 0). When x = 0, y = (1/12)(0)² = 0, so the point (0, 0) lies on the parabola. Furthermore, the axis of symmetry is the y-axis, as the equation is symmetric with respect to x (i.e., replacing x with -x does not change the equation).

General Equation of a Parabola

It's helpful to understand the general form of a parabola's equation. The standard form of a parabola with a vertical axis of symmetry and vertex at the origin is:

y = (1/(4p))x²

where 'p' is the distance from the vertex to the focus (and also the distance from the vertex to the directrix). In our example, the distance from the vertex (0, 0) to the focus (0, 3) is 3, so p = 3. Plugging this into the general equation, we get:

y = (1/(4*3))x²
y = (1/12)x²

This confirms our derived equation and provides a useful framework for solving similar problems.

Choosing the Correct Option

Now, let's revisit the original question. We were given the following options:

a. y = (1/12)x² b. y = (1/3)x² c. y = 1/(12x²) d. y = 1/(3x²)

Based on our derivation, the correct equation for the parabola with a focus at (0, 3) and directrix y = -3 is:

a. y = (1/12)x²

Key Takeaways

This exercise demonstrates a powerful method for finding the equation of a parabola using its fundamental definition. Here are some key takeaways:

  • Definition is Key: The definition of a parabola as the set of points equidistant from the focus and directrix is the foundation for deriving its equation.
  • Distance Formulas: The distance formula and the formula for the distance between a point and a line are essential tools.
  • Standard Form: Understanding the standard form of a parabola's equation helps in recognizing and interpreting the results.
  • Visualization: Visualizing the focus, directrix, and vertex on a coordinate plane aids in understanding the parabola's orientation and shape.

Further Exploration

This problem provides a solid foundation for understanding parabolas. To further your understanding, consider exploring the following:

  • Parabolas with Horizontal Axes: Investigate parabolas that open to the left or right, where the directrix is a vertical line.
  • Parabolas with Vertices Not at the Origin: Explore how the equation changes when the vertex is shifted away from the origin.
  • Applications of Parabolas: Research real-world applications of parabolas, such as in satellite dishes, reflectors, and projectile motion.

By continuing to explore these concepts, you'll deepen your understanding of parabolas and their significance in mathematics and beyond.

In conclusion, finding the equation of a parabola involves applying its definition, utilizing distance formulas, and simplifying algebraic expressions. By understanding these steps and the underlying concepts, you can confidently tackle a wide range of parabola-related problems. The example we worked through, finding the equation of a parabola with a focus at (0, 3) and directrix y = -3, illustrates this process effectively, providing a clear path to the correct solution: y = (1/12)x². Remember to always visualize the geometry and understand the key properties of the parabola, such as the focus, directrix, vertex, and axis of symmetry. This will not only help you solve problems but also deepen your appreciation for this fascinating conic section.