Parabola Analysis Determining Axis And Vertex Of 16x² - 24xy + 9y² - 104x - 172y + 44 = 0
Introduction
In the realm of conic sections, the parabola stands out as a fascinating curve with a rich set of properties and applications. Its unique shape, formed by the intersection of a cone and a plane parallel to one of its sides, has captivated mathematicians and scientists for centuries. Understanding the equation of a parabola and its key features, such as its axis and vertex, is crucial in various fields, from optics and mechanics to engineering and computer graphics. In this article, we embark on a comprehensive journey to analyze the equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0, demonstrating that it represents a parabola and meticulously determining the equation of its axis and the coordinates of its vertex. This exploration will not only solidify our understanding of parabolas but also highlight the power of algebraic techniques in deciphering geometric forms. We will delve into the intricacies of conic section identification, rotation of axes, and the derivation of standard parabolic forms, providing a detailed roadmap for readers to follow and comprehend. The equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0 presents a general quadratic form, which could represent a variety of conic sections, including ellipses, hyperbolas, or parabolas. Our initial task is to establish definitively that this equation corresponds specifically to a parabola. This involves examining the discriminant of the equation, a crucial indicator that unveils the nature of the conic section. By calculating the discriminant and confirming its value, we can confidently classify the given equation as representing a parabola. This foundational step sets the stage for the subsequent analysis, where we delve into the intricacies of determining the axis and vertex. The axis of a parabola is a line of symmetry that divides the parabola into two congruent halves. Finding the equation of the axis involves a series of transformations and algebraic manipulations, ultimately revealing the line about which the parabola is perfectly symmetrical. The vertex, on the other hand, is the point where the parabola changes direction, representing either the minimum or maximum point on the curve. Determining the vertex's coordinates requires a careful analysis of the transformed equation, leading us to pinpoint the exact location of this critical point. Through this detailed exploration, we aim to provide a clear and comprehensive understanding of the parabola represented by the given equation, empowering readers with the knowledge and skills to analyze similar conic sections in the future.
Proving the Equation Represents a Parabola
To demonstrate that the equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0 represents a parabola, we need to analyze the general form of a conic section equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0. The discriminant, given by Δ = B² - 4AC, plays a crucial role in identifying the type of conic section. If Δ = 0, the equation represents a parabola; if Δ < 0, it represents an ellipse; and if Δ > 0, it represents a hyperbola. In our given equation, A = 16, B = -24, and C = 9. Therefore, the discriminant is calculated as follows:
Δ = (-24)² - 4(16)(9) = 576 - 576 = 0
Since the discriminant Δ = 0, we can definitively conclude that the equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0 represents a parabola. This crucial step confirms our initial hypothesis and paves the way for further analysis to determine the parabola's key characteristics, such as its axis and vertex. The discriminant serves as a powerful tool in conic section analysis, providing a concise way to classify the curve represented by a general quadratic equation. By simply calculating the value of B² - 4AC, we can immediately ascertain whether the equation corresponds to a parabola, ellipse, or hyperbola. In the case of a parabola, the discriminant's value of zero signifies a unique relationship between the coefficients of the quadratic terms, indicating a specific geometric configuration that results in the characteristic parabolic shape. The next stage in our analysis involves finding the equation of the axis and the coordinates of the vertex. These elements provide a deeper understanding of the parabola's orientation and position in the coordinate plane. The axis of a parabola is a line of symmetry that divides the curve into two mirror-image halves, while the vertex represents the point where the parabola changes direction, marking either its minimum or maximum value. Determining these features requires a series of algebraic manipulations and transformations, which we will explore in detail in the subsequent sections. The confirmation that the equation represents a parabola is a foundational step, allowing us to proceed with confidence in our pursuit of its specific characteristics. This process exemplifies the power of mathematical tools in unraveling the geometric properties hidden within algebraic expressions, highlighting the interconnectedness of different branches of mathematics. As we move forward, we will employ techniques such as rotation of axes and completing the square to simplify the equation and reveal the parabola's underlying structure, ultimately leading us to the precise determination of its axis and vertex.
Finding the Equation of the Axis
The equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0 contains an xy term, indicating that the parabola is rotated with respect to the standard coordinate axes. To find the equation of the axis, we first need to eliminate the xy term by rotating the coordinate axes. The angle of rotation, θ, can be found using the formula:
tan(2θ) = B / (A - C)
In our case, A = 16, B = -24, and C = 9. Therefore:
tan(2θ) = -24 / (16 - 9) = -24 / 7
To find θ, we can use the inverse tangent function. However, it's more convenient to find cos(2θ) and sin(2θ) directly using the trigonometric identity tan²(2θ) + 1 = sec²(2θ). First, we find sec²(2θ):
sec²(2θ) = (-24/7)² + 1 = 576/49 + 1 = 625/49
Taking the reciprocal, we get cos²(2θ) = 49/625. Since tan(2θ) < 0 and we want 0 < 2θ < π, we have cos(2θ) < 0. Thus:
cos(2θ) = -7/25
Using the identity sin²(2θ) + cos²(2θ) = 1, we find:
sin²(2θ) = 1 - (-7/25)² = 1 - 49/625 = 576/625
Since 0 < 2θ < π, we have sin(2θ) > 0. Thus:
sin(2θ) = 24/25
Now, we can find sin(θ) and cos(θ) using the half-angle formulas:
cos(θ) = √((1 + cos(2θ)) / 2) = √((1 - 7/25) / 2) = √(18/50) = √(9/25) = 3/5
sin(θ) = √((1 - cos(2θ)) / 2) = √((1 + 7/25) / 2) = √(32/50) = √(16/25) = 4/5
The rotation equations are:
x = x'cos(θ) - y'sin(θ) = (3/5)x' - (4/5)y'
y = x'sin(θ) + y'cos(θ) = (4/5)x' + (3/5)y'
Substituting these into the original equation and simplifying, we get:
16((3/5)x' - (4/5)y')² - 24((3/5)x' - (4/5)y')((4/5)x' + (3/5)y') + 9((4/5)x' + (3/5)y')² - 104((3/5)x' - (4/5)y') - 172((4/5)x' + (3/5)y') + 44 = 0
After extensive algebraic simplification (which is a tedious but necessary step), the equation transforms into:
25y'² - 200x' - 20y' + 44 = 0
Completing the square for the y' terms:
25(y'² - (4/5)y') = 200x' - 44
25(y'² - (4/5)y' + (2/5)²) = 200x' - 44 + 25(4/25)
25(y' - 2/5)² = 200x' - 40
(y' - 2/5)² = 8(x' - 1/5)
This is the standard form of a parabola, (y - k)² = 4p(x - h), where the vertex is (h, k). The axis of this parabola is parallel to the x'-axis and has the equation y' = 2/5. To find the equation of the axis in the original xy-coordinates, we substitute y' back using the rotation equation:
y' = (5y - 4x) / 5
(5y - 4x) / 5 = 2/5
5y - 4x = 2
Therefore, the equation of the axis of the parabola is 4x - 5y + 2 = 0. This equation represents the line of symmetry that bisects the parabola, a crucial element in understanding its geometric structure.
Determining the Vertex
Now that we have the equation of the parabola in the rotated coordinate system, (y' - 2/5)² = 8(x' - 1/5), we can easily identify the vertex. In the standard form of a parabola, (y - k)² = 4p(x - h), the vertex is located at the point (h, k). Comparing this with our equation, we see that the vertex in the rotated coordinate system is (x', y') = (1/5, 2/5). To find the coordinates of the vertex in the original xy-coordinate system, we use the rotation equations:
x = x'cos(θ) - y'sin(θ)
y = x'sin(θ) + y'cos(θ)
Substituting x' = 1/5, y' = 2/5, cos(θ) = 3/5, and sin(θ) = 4/5, we get:
x = (1/5)(3/5) - (2/5)(4/5) = 3/25 - 8/25 = -5/25 = -1/5
y = (1/5)(4/5) + (2/5)(3/5) = 4/25 + 6/25 = 10/25 = 2/5
Therefore, the vertex of the parabola in the original xy-coordinates is (-1/5, 2/5). This point represents the extreme point of the parabola, either the minimum or maximum, and is a key feature in understanding its shape and position. The vertex, along with the axis, provides a complete picture of the parabola's orientation and location in the coordinate plane. The process of finding the vertex involved transforming the equation into its standard form, identifying the vertex in the rotated coordinate system, and then converting it back to the original coordinate system using the rotation equations. This multi-step process highlights the power of coordinate transformations in simplifying complex geometric problems and extracting key information about the curves they represent. The vertex, as the turning point of the parabola, plays a significant role in various applications, from optics and antenna design to projectile motion and curve fitting. Its precise location is crucial in understanding the parabola's behavior and utilizing its properties effectively.
Conclusion
In this detailed analysis, we have successfully demonstrated that the equation 16x² - 24xy + 9y² - 104x - 172y + 44 = 0 represents a parabola. We achieved this by calculating the discriminant, Δ = B² - 4AC, and verifying that it equals zero. Furthermore, we have meticulously determined the equation of the axis of the parabola to be 4x - 5y + 2 = 0 and identified the vertex as the point (-1/5, 2/5). This journey involved a series of algebraic manipulations, including the rotation of axes to eliminate the xy term and completing the square to transform the equation into its standard parabolic form. The process showcased the power of analytical geometry in deciphering the properties of conic sections and provided a comprehensive understanding of the parabola represented by the given equation. The techniques employed, such as discriminant analysis, rotation of axes, and completing the square, are fundamental tools in the study of conic sections and have broad applications in various fields of mathematics, science, and engineering. By understanding these methods, readers can confidently analyze and interpret similar quadratic equations, extracting valuable geometric information and applying it to real-world problems. The parabola, as one of the fundamental conic sections, plays a crucial role in numerous applications, from the design of parabolic reflectors in telescopes and antennas to the modeling of projectile motion in physics. Its unique shape and properties make it a versatile tool for solving a wide range of problems. The detailed analysis presented in this article not only provides a specific solution for the given equation but also offers a general framework for analyzing other conic sections, empowering readers to explore the fascinating world of geometric curves and their applications. The ability to identify, characterize, and manipulate conic sections is a valuable skill for anyone pursuing advanced studies in mathematics, science, or engineering. This exploration serves as a stepping stone towards a deeper understanding of the mathematical principles that govern the shapes and forms we observe in the world around us. From the graceful arc of a bridge to the precise trajectory of a projectile, the parabola continues to be a testament to the power and elegance of mathematical concepts.
