Equation Of A Parabola Focus (4, 6) And Directrix Y = -6
Finding the equation of a parabola given its focus and directrix is a fundamental concept in conic sections. In this comprehensive guide, we will delve into the intricacies of parabolas, exploring their defining properties, and derive the equation of a parabola with a focus at (4, 6) and a directrix of y = -6. Let's embark on this mathematical journey to understand the underlying principles and arrive at the correct equation.
Understanding the Parabola: A Geometric Definition
At its core, a parabola is a symmetrical, U-shaped curve defined as the set of all points that are equidistant from a fixed point called the focus and a fixed line called the directrix. This fundamental property forms the basis for deriving the equation of a parabola. The focus, a crucial element, lies inside the curve, while the directrix is a line that resides outside the curve. The vertex, another significant point, is the midpoint between the focus and the directrix. Understanding these key components is crucial to grasp the essence of a parabola and its equation.
To visualize this, imagine a point moving in a plane such that its distance from the focus is always equal to its distance from the directrix. The path traced by this point forms the parabola. This equidistance property is the heart of the parabola's definition and is essential for deriving its equation. In essence, the parabola is a geometric representation of this balance between distances, creating its unique and characteristic shape.
Key Components: Focus, Directrix, and Vertex
To fully grasp the equation of a parabola, we need to understand the roles of its key components: the focus, the directrix, and the vertex. The focus is a fixed point that defines the curvature of the parabola. The closer the focus is to the vertex, the tighter the curve. Conversely, the farther the focus is from the vertex, the wider the curve. The directrix is a fixed line that, together with the focus, determines the shape and position of the parabola. It acts as a guideline, ensuring that every point on the parabola maintains an equal distance from both the focus and the directrix. The vertex, as mentioned earlier, is the turning point of the parabola and lies exactly halfway between the focus and the directrix. It's the point where the parabola changes direction and is a crucial reference point for determining the parabola's equation.
These three components work in harmony to define the unique characteristics of a parabola. By understanding their relationships, we can effectively analyze and derive the equation of any parabola, given its focus and directrix.
Deriving the Equation: A Step-by-Step Approach
Now, let's translate this geometric definition into an algebraic equation. Consider a point (x, y) on the parabola. According to the definition, the distance from (x, y) to the focus must equal the distance from (x, y) to the directrix. This equidistance property is the key to unlocking the equation of the parabola.
- Distance to the Focus: We employ the distance formula to calculate the distance between the point (x, y) and the focus (4, 6). The distance formula, derived from the Pythagorean theorem, states that the distance between two points (x1, y1) and (x2, y2) is √((x2 - x1)² + (y2 - y1)²). Applying this to our scenario, the distance from (x, y) to (4, 6) is √((x - 4)² + (y - 6)²).
- Distance to the Directrix: The distance from the point (x, y) to the directrix y = -6 is the vertical distance between the point and the line. This distance is simply the absolute value of the difference in their y-coordinates, which is |y - (-6)| or |y + 6|.
- Equating the Distances: According to the definition of a parabola, these two distances must be equal. Therefore, we set the distance to the focus equal to the distance to the directrix: √((x - 4)² + (y - 6)²) = |y + 6|.
- Simplifying the Equation: To eliminate the square root and absolute value, we square both sides of the equation: (x - 4)² + (y - 6)² = (y + 6)². Expanding the squared terms, we get (x - 4)² + y² - 12y + 36 = y² + 12y + 36. Notice that the y² and 36 terms cancel out on both sides. This leaves us with (x - 4)² - 12y = 12y. Adding 12y to both sides, we arrive at (x - 4)² = 24y.
Therefore, the equation of the parabola with a focus at (4, 6) and a directrix of y = -6 is (x - 4)² = 24y. This equation represents the locus of all points that satisfy the equidistance condition, forming the parabolic curve.
Analyzing the Equation: Understanding the Form
The equation we derived, (x - 4)² = 24y, is in the standard form of a parabola that opens upwards. This form provides valuable insights into the parabola's characteristics. The general form of a parabola opening upwards or downwards is (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance between the vertex and the focus (and also the distance between the vertex and the directrix).
In our equation, (x - 4)² = 24y, we can identify the vertex as (4, 0). This is because the equation can be rewritten as (x - 4)² = 4(6)(y - 0), matching the standard form. The value of p is 6, which confirms that the distance between the vertex (4, 0) and the focus (4, 6) is indeed 6 units. Similarly, the distance between the vertex (4, 0) and the directrix y = -6 is also 6 units. This consistency validates our derived equation and reinforces the equidistance property of parabolas.
Moreover, the positive coefficient 24 in front of the y term indicates that the parabola opens upwards. If the coefficient were negative, the parabola would open downwards. This directional information is crucial for visualizing the parabola and understanding its orientation in the coordinate plane. By analyzing the equation in this standard form, we gain a deeper understanding of the parabola's properties and its relationship to the focus, directrix, and vertex.
Choosing the Correct Option: Matching the Derived Equation
Having derived the equation (x - 4)² = 24y, we can now confidently identify the correct option from the given choices. Let's revisit the options:
- (x - 4)² = 24y
- (x - 4)² = (1/24)y
- (x - 4)² = -24y
- (x - 4)² = -(1/24)y
Clearly, the first option, (x - 4)² = 24y, matches our derived equation. Therefore, this is the correct representation of the parabola with a focus at (4, 6) and a directrix of y = -6. The other options either have incorrect coefficients or signs, indicating parabolas with different shapes or orientations.
By systematically deriving the equation and comparing it with the given options, we can confidently select the correct answer. This process highlights the importance of understanding the underlying principles of parabolas and their equations.
Conclusion: Mastering the Parabola
In this comprehensive exploration, we have successfully derived the equation of a parabola with a focus at (4, 6) and a directrix of y = -6. We began by understanding the geometric definition of a parabola, emphasizing the crucial equidistance property between the focus, directrix, and points on the curve. We then translated this definition into an algebraic equation using the distance formula and simplified it to obtain the standard form (x - 4)² = 24y. Analyzing this equation, we identified the vertex, the distance between the vertex and focus, and the direction in which the parabola opens.
Finally, by comparing our derived equation with the given options, we confidently selected the correct representation of the parabola. This journey through the properties and equation of a parabola demonstrates the power of combining geometric understanding with algebraic manipulation. By mastering these concepts, you can confidently tackle a wide range of problems involving parabolas and conic sections.
