Determining The Value Of C For A Parabola In The Equation 2x² + Cy² - 12x - 10y = 96

Introduction

In the realm of conic sections, the parabola stands as a captivating curve, defined by its unique properties and elegant equation. Understanding the conditions that dictate whether a given equation represents a parabola is crucial for mastering this fundamental concept in mathematics. This article delves into the intricacies of the equation 2x² + Cy² - 12x - 10y = 96, aiming to determine the value of C that transforms its graph into a parabola. By exploring the general form of conic sections and analyzing the specific characteristics of parabolas, we will unravel the mystery behind this equation and uncover the precise value of C that satisfies the parabolic condition.

The General Equation of Conic Sections

To embark on our quest, it's essential to first grasp the general equation that governs all conic sections. This equation, a powerful tool in the world of mathematics, takes the form:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Here, A, B, C, D, E, and F are constants that dictate the shape and orientation of the conic section. By carefully manipulating these constants, we can conjure a diverse array of curves, including circles, ellipses, hyperbolas, and, of course, parabolas. Each conic section possesses a unique set of characteristics, making them indispensable in various fields, from optics and astronomy to engineering and architecture.

Parabolas: A Distinctive Conic Section

Among the conic sections, the parabola holds a special place, distinguished by its characteristic U-shape. This elegant curve is defined as the set of all points that are equidistant to a fixed point, known as the focus, and a fixed line, known as the directrix. The parabola's unique geometry gives rise to its distinctive properties, making it a cornerstone in various scientific and technological applications.

The Parabolic Condition: A Key to Unveiling Parabolas

The key to identifying a parabola within the realm of conic sections lies in a specific condition. In the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, a parabola emerges when the discriminant, B² - 4AC, equals zero. This condition, a mathematical fingerprint, allows us to distinguish parabolas from their conic brethren, ensuring we can accurately identify and analyze these captivating curves.

Analyzing the Equation 2x² + Cy² - 12x - 10y = 96

Now, let's turn our attention to the equation at hand: 2x² + Cy² - 12x - 10y = 96. To determine the value of C that transforms this equation into a parabola, we must apply the parabolic condition. First, we need to rearrange the equation into the general form of a conic section:

2x² + Cy² - 12x - 10y - 96 = 0

Comparing this equation to the general form, we can identify the coefficients:

• A = 2
• B = 0
• C = C

Applying the Parabolic Condition

With the coefficients identified, we can now apply the parabolic condition, B² - 4AC = 0. Substituting the values, we get:

0² - 4 * 2 * C = 0

Simplifying the equation, we have:

-8C = 0

Solving for C, we find:

C = 0

Therefore, the value of C that transforms the equation 2x² + Cy² - 12x - 10y = 96 into a parabola is 0.

Deeper Dive into Parabolas

Now that we've determined the value of C, let's delve deeper into the characteristics of parabolas and explore their significance in various fields.

Key Features of a Parabola

A parabola is defined by several key features, each contributing to its unique shape and properties:

• Focus: A fixed point within the parabola.
• Directrix: A fixed line outside the parabola.
• Vertex: The point on the parabola closest to both the focus and the directrix.
• Axis of Symmetry: A line passing through the focus and vertex, dividing the parabola into two symmetrical halves.

Applications of Parabolas

Parabolas find widespread applications in various fields, leveraging their unique properties to solve real-world problems:

• Optics: Parabolic mirrors and reflectors are used in telescopes, satellite dishes, and headlights, focusing light or radio waves to a single point.
• Engineering: Parabolic arches provide structural support in bridges and buildings, distributing weight evenly.
• Sports: The trajectory of a ball thrown or kicked often approximates a parabolic path, influenced by gravity and air resistance.

Conclusion

In this exploration of the equation 2x² + Cy² - 12x - 10y = 96, we successfully determined that the value of C must be 0 for the graph to represent a parabola. By understanding the general equation of conic sections and applying the parabolic condition, we unraveled the mystery behind this equation and gained a deeper appreciation for the unique characteristics of parabolas.

Parabolas, with their elegant U-shape and remarkable properties, continue to captivate mathematicians and scientists alike. From their role in focusing light to their structural applications in engineering, parabolas stand as a testament to the power of mathematical concepts in shaping our world. This journey into the realm of parabolas has not only enhanced our understanding of conic sections but has also highlighted the interconnectedness of mathematics and the world around us.

Further Exploration

To further expand your understanding of parabolas, consider exploring the following topics:

• Derivation of the Parabola Equation: Delve into the mathematical derivation of the standard equation of a parabola.
• Parabolic Reflectors: Investigate the principles behind parabolic mirrors and their applications in telescopes and satellite dishes.
• Applications in Physics: Explore how parabolas describe projectile motion and other physical phenomena.
• Conic Sections in Nature: Discover real-world examples of conic sections in nature, from planetary orbits to the shape of certain flowers.

By continuing your exploration of parabolas and conic sections, you'll unlock a deeper appreciation for the elegance and power of mathematics in shaping our understanding of the world.

This article serves as a stepping stone in your mathematical journey, encouraging you to delve further into the fascinating world of conic sections and their applications. Remember, the quest for knowledge is a never-ending adventure, and the more you explore, the more you'll discover the beauty and interconnectedness of mathematics.