Standard Form Of A Parabola Equation Focus (8,0) Directrix X=-8

The question at hand delves into the fundamental properties of parabolas and their representation in standard form. Understanding the relationship between a parabola's focus, directrix, and its equation is crucial for solving this problem. We will meticulously explore the definition of a parabola, derive its standard equation based on the given focus and directrix, and then confidently select the correct answer from the provided options. This comprehensive exploration will solidify your understanding of parabolas and their equations.

Defining the Parabola: Focus, Directrix, and the Locus of Points

At its heart, a parabola is defined as the locus of all points that are equidistant to a fixed point, known as the focus, and a fixed line, known as the directrix. This definition is the cornerstone for understanding and deriving the equation of a parabola. The focus is a crucial element, dictating the curvature and orientation of the parabola. The directrix, on the other hand, acts as a boundary line, ensuring that the parabola opens away from it. Imagine a point tracing a path such that its distance to the focus is always equal to its distance to the directrix – the path it traces is a parabola.

The significance of this definition lies in its ability to translate the geometric concept of a parabola into an algebraic equation. By using the distance formula and the definition of a parabola, we can mathematically express the relationship between the coordinates of a point on the parabola and the focus and directrix. This algebraic representation is what we refer to as the equation of the parabola. The standard form of this equation provides a concise and informative way to represent the parabola, highlighting its key characteristics such as its vertex, focus, and direction of opening.

Deriving the Standard Equation: A Step-by-Step Approach

To derive the standard equation of the parabola given a focus of (8,0) and a directrix of x = -8, we start by considering a general point (x, y) on the parabola. According to the definition of a parabola, the distance from this point to the focus must be equal to the distance from this point to the directrix.

The distance between the point (x, y) and the focus (8, 0) can be calculated using the distance formula:

√((x - 8)² + (y - 0)²) = √((x - 8)² + y²)

The distance between the point (x, y) and the directrix x = -8 is the perpendicular distance, which is simply the absolute difference between the x-coordinate of the point and the x-coordinate of the directrix:

|x - (-8)| = |x + 8|

Now, we equate these two distances, reflecting the fundamental definition of a parabola:

√((x - 8)² + y²) = |x + 8|

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

(x - 8)² + y² = (x + 8)²

Expanding the squared terms, we get:

x² - 16x + 64 + y² = x² + 16x + 64

Simplifying the equation by canceling out common terms (x² and 64) and rearranging, we obtain:

y² = 32x

This is the standard form equation of the parabola with the given focus and directrix. It clearly shows the relationship between the y-coordinate and the x-coordinate of any point on the parabola. The equation is in the form y² = 4ax, where 'a' is the distance from the vertex to the focus (and also the distance from the vertex to the directrix). In this case, 4a = 32, so a = 8. This confirms that the focus is indeed at (8, 0) and the directrix is at x = -8, as given in the problem.

Analyzing the Equation: Key Features of the Parabola

Now that we have derived the standard equation y² = 32x, we can analyze it to understand the key features of the parabola. The equation is in the form y² = 4ax, which signifies a parabola that opens to the right. The vertex of the parabola is at the origin (0, 0), which is the midpoint between the focus and the directrix. The axis of symmetry is the x-axis (y = 0), which is the line passing through the focus and perpendicular to the directrix.

The value of 'a', which we found to be 8, represents the distance from the vertex to the focus and also the distance from the vertex to the directrix. This parameter is crucial in determining the shape and size of the parabola. A larger value of 'a' indicates a wider parabola, while a smaller value indicates a narrower parabola. The focus, located at (a, 0), is the point where all incoming rays parallel to the axis of symmetry converge after reflection from the parabolic surface. This property makes parabolas essential in applications such as satellite dishes and reflecting telescopes.

The directrix, given by the equation x = -a, is a vertical line that lies 'a' units to the left of the vertex. It acts as a boundary line, ensuring that the parabola opens away from it. The distance between any point on the parabola and the focus is always equal to the perpendicular distance between that point and the directrix, as we established earlier. Understanding these features allows us to visualize the parabola and its orientation in the coordinate plane, providing a deeper understanding of its properties and behavior.

Selecting the Correct Answer: Matching the Derived Equation

Having derived the standard equation y² = 32x, we can now confidently select the correct answer from the given options. The options were:

A. y² = -8x B. y² = 8x C. y² = 32x D. y² = -32x

By comparing our derived equation with the options, it is clear that option C, y² = 32x, is the correct answer. The other options have either the wrong coefficient for x or the wrong sign, indicating a different parabola or a parabola opening in a different direction. Option A and D represent parabolas that open to the left, while option B represents a parabola with a different curvature.

This exercise highlights the importance of a systematic approach to solving mathematical problems. By understanding the definition of a parabola, deriving its equation from the given information, and analyzing the resulting equation, we were able to confidently arrive at the correct answer. This approach not only provides the solution but also deepens our understanding of the underlying concepts and principles.

Conclusion: Mastering the Parabola and its Equation

In conclusion, the standard form of the equation of a parabola with a focus of (8,0) and directrix x=-8 is y² = 32x. This was determined by applying the definition of a parabola as the locus of points equidistant to the focus and directrix, deriving the equation using the distance formula, and simplifying the result. Understanding the relationship between the focus, directrix, and the equation of a parabola is fundamental to solving problems related to conic sections and their applications.

The process of deriving the equation involved several key steps, including understanding the definition of a parabola, applying the distance formula, and algebraic manipulation. Each step is crucial in arriving at the correct solution. Furthermore, analyzing the derived equation provides valuable insights into the characteristics of the parabola, such as its vertex, axis of symmetry, and direction of opening. This deeper understanding enhances our ability to solve similar problems and apply the concepts to real-world scenarios.

By mastering the standard form of the equation of a parabola, we gain a powerful tool for analyzing and understanding these essential geometric shapes. Parabolas appear in various fields, from optics and antenna design to projectile motion and bridge construction. A strong grasp of their properties and equations is essential for success in mathematics, physics, and engineering. This comprehensive guide has provided a step-by-step approach to understanding the parabola, its equation, and its significance, empowering you to tackle future challenges with confidence.