Hyperbola Comparison Analyzing $\frac{x^2}{6^2}-\frac{y^2}{8^2}=1$ And $\frac{x^2}{8^2}-\frac{y^2}{6^2}=1$

Jul 14, 2025 by ADMIN 111 views

$Exploring Hyperbolas A Comparative Analysis of ${\frac{x^2}{6^2}-\frac{y^2}{8^2}=1}$ and ${\frac{x^2}{8^2}-\frac{y^2}{6^2}=1}$$

The fascinating world of conic sections introduces us to hyperbolas, curves defined by a constant difference of distances from two fixed points, known as foci. In this comprehensive exploration, we will delve into a comparative analysis of two hyperbolas represented by the equations ${\frac{x^2}{6^2}-\frac{y^2}{8^2}=1}$ and ${\frac{x^2}{8^2}-\frac{y^2}{6^2}=1}$ . Our primary objective is to identify a true statement that accurately compares the graphs of these two hyperbolic equations. To achieve this, we will meticulously examine their key characteristics, including their foci, transverse axes, conjugate axes, and overall orientation. By understanding these fundamental properties, we can gain a deeper appreciation for the unique nature of each hyperbola and their relationships to one another. So, let's embark on this mathematical journey and unravel the intricacies of these hyperbolic curves.

Understanding the Standard Form of a Hyperbola

Before we embark on a detailed comparison of the given hyperbolas, it's crucial to grasp the standard form equations that govern their behavior. Hyperbolas, as conic sections, exhibit distinct equations that dictate their shape and orientation within the Cartesian plane. The general standard forms for hyperbolas centered at the origin are:

Horizontal Hyperbola: ${\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1}$
Vertical Hyperbola: ${\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$

In these equations, 'a' represents the distance from the center of the hyperbola to a vertex along the transverse axis, while 'b' denotes the distance from the center to a vertex along the conjugate axis. The transverse axis is the axis that passes through the foci and vertices, dictating the hyperbola's primary direction of opening. Conversely, the conjugate axis is perpendicular to the transverse axis and influences the hyperbola's overall shape. The orientation of the hyperbola, whether horizontal or vertical, is determined by which term (x² or y²) is positive in the equation. A positive x² term indicates a horizontal hyperbola, whereas a positive y² term signifies a vertical hyperbola.

The parameters 'a' and 'b' are instrumental in determining other critical features of the hyperbola, including the location of the foci and the equations of the asymptotes. The foci, the two fixed points that define the hyperbola, lie on the transverse axis, a distance of 'c' units from the center. This distance 'c' is related to 'a' and 'b' by the equation ${c^2 = a^2 + b^2}$ . The asymptotes, which are lines that the hyperbola approaches as it extends to infinity, intersect at the center of the hyperbola. Their slopes are given by ±(b/a) for a horizontal hyperbola and ±(a/b) for a vertical hyperbola.

With a solid understanding of the standard form equations and their parameters, we are now well-equipped to analyze the specific hyperbolas presented in our problem and make meaningful comparisons.

Analyzing the First Hyperbola: ${\frac{x^2}{6^2}-\frac{y^2}{8^2}=1}$

Let's turn our attention to the first hyperbola, described by the equation ${\frac{x^2}{6^2}-\frac{y^2}{8^2}=1}$ . This equation is in the standard form for a horizontal hyperbola, where the x² term is positive. By carefully examining the equation, we can extract key information about its characteristics. The denominator under the x² term, 6², tells us that a = 6. This value represents the distance from the center of the hyperbola to each vertex along the transverse axis. Since it's a horizontal hyperbola, the vertices lie on the x-axis, located at (±6, 0). Similarly, the denominator under the y² term, 8², reveals that b = 8. This value represents the distance from the center to the vertices along the conjugate axis, which in this case is the y-axis.

To determine the location of the foci, we need to calculate the distance 'c' using the relationship ${c^2 = a^2 + b^2}$ . Substituting the values of a and b, we get ${c^2 = 6^2 + 8^2 = 36 + 64 = 100}$ . Taking the square root of both sides, we find that c = 10. Therefore, the foci of this hyperbola are located at (±10, 0), lying on the x-axis, a distance of 10 units from the center.

The transverse axis, which connects the vertices and passes through the foci, has a length of 2a. In this case, the length of the transverse axis is 2 * 6 = 12 units. This axis stretches horizontally between the vertices at (-6, 0) and (6, 0). The conjugate axis, perpendicular to the transverse axis, has a length of 2b. Here, the length of the conjugate axis is 2 * 8 = 16 units. This axis extends vertically, influencing the hyperbola's overall shape.

Finally, let's consider the asymptotes of this hyperbola. Asymptotes are lines that the hyperbola approaches as it extends infinitely. For a horizontal hyperbola, the slopes of the asymptotes are given by ±(b/a). In our case, the slopes are ±(8/6) = ±(4/3). Since the hyperbola is centered at the origin, the equations of the asymptotes are y = (4/3)x and y = -(4/3)x. These lines provide a framework for the hyperbola's branches, guiding their paths as they extend away from the center.

Analyzing the Second Hyperbola: ${\frac{x^2}{8^2}-\frac{y^2}{6^2}=1}$

Now, let's shift our focus to the second hyperbola, defined by the equation ${\frac{x^2}{8^2}-\frac{y^2}{6^2}=1}$ . Similar to the first hyperbola, this equation also represents a horizontal hyperbola since the x² term is positive. However, a crucial difference lies in the values of 'a' and 'b'. In this case, the denominator under the x² term, 8², indicates that a = 8. This value signifies the distance from the center to each vertex along the transverse axis. As it's a horizontal hyperbola, the vertices are situated on the x-axis at coordinates (±8, 0).

The denominator under the y² term, 6², reveals that b = 6. This value represents the distance from the center to the vertices along the conjugate axis, which is the y-axis in this instance. To determine the foci's location, we again utilize the relationship ${c^2 = a^2 + b^2}$ . Substituting the values of a and b, we get ${c^2 = 8^2 + 6^2 = 64 + 36 = 100}$ . Taking the square root of both sides, we find that c = 10. Therefore, the foci of this hyperbola are located at (±10, 0), mirroring the foci of the first hyperbola.

The transverse axis, connecting the vertices and passing through the foci, has a length of 2a. For this hyperbola, the length of the transverse axis is 2 * 8 = 16 units, a notable difference from the first hyperbola. This axis extends horizontally between the vertices at (-8, 0) and (8, 0). The conjugate axis, perpendicular to the transverse axis, has a length of 2b, which is 2 * 6 = 12 units. Again, this length differs from the first hyperbola.

Turning to the asymptotes, their slopes are determined by ±(b/a) for a horizontal hyperbola. In this case, the slopes are ±(6/8) = ±(3/4). Since the hyperbola is centered at the origin, the equations of the asymptotes are y = (3/4)x and y = -(3/4)x. These lines, while different from those of the first hyperbola, serve the same purpose of guiding the branches of the hyperbola as they extend away from the center.

Comparative Analysis and Identifying the True Statement

Having meticulously analyzed each hyperbola individually, we are now primed to engage in a comparative analysis, seeking to pinpoint the true statement that accurately contrasts their graphs. Let's systematically compare their key features:

Foci: As we calculated, both hyperbolas share the same foci, located at (±10, 0). This immediately suggests that option A, which posits that the foci of both graphs are the same points, is a strong contender for the true statement.
Transverse Axis: The first hyperbola has a transverse axis length of 12 units, while the second hyperbola boasts a transverse axis length of 16 units. This disparity indicates that option B, which claims that the lengths of both transverse axes are the same, is incorrect.
Orientation: Both hyperbolas are horizontal, as evidenced by the positive x² term in their equations.
Asymptotes: The asymptotes of the first hyperbola have slopes of ±(4/3), while the asymptotes of the second hyperbola have slopes of ±(3/4). This difference in slopes implies that the hyperbolas will approach different lines as they extend infinitely.

Based on our comparative analysis, the evidence overwhelmingly supports the conclusion that the foci of both graphs are indeed the same points. The other options presented in the problem can be confidently ruled out due to the differences in transverse axis lengths and asymptote slopes.

Therefore, the true statement comparing the graphs of ${\frac{x^2}{6^2}-\frac{y^2}{8^2}=1}$ and ${\frac{x^2}{8^2}-\frac{y^2}{6^2}=1}$ is:

A. The foci of both graphs are the same points.

Conclusion

In this comprehensive exploration, we embarked on a journey to compare two hyperbolas, meticulously analyzing their equations and extracting key characteristics. By understanding the standard form of a hyperbola, we were able to determine the values of 'a', 'b', and 'c', which in turn allowed us to locate the vertices, foci, and asymptotes. Our comparative analysis revealed that while the hyperbolas differed in their transverse axis lengths and asymptote slopes, they shared the same foci. This led us to the identification of the true statement, solidifying our understanding of these fascinating conic sections. The exploration of hyperbolas not only enhances our mathematical knowledge but also cultivates our analytical and problem-solving skills. By dissecting equations, comparing properties, and drawing logical conclusions, we empower ourselves to tackle complex mathematical challenges with confidence and precision. The world of conic sections, with its intricate curves and geometric relationships, offers a rich landscape for mathematical exploration, inviting us to delve deeper into the beauty and elegance of mathematical principles.