Finding Center, Eccentricity, Vertices, Foci, Directrix, And Axes Of A Hyperbola
In the realm of conic sections, the hyperbola stands out as a fascinating curve with its distinctive two-branched shape. Understanding its properties and parameters is crucial in various fields, from physics and astronomy to engineering and computer graphics. This comprehensive guide aims to dissect the hyperbola, providing a step-by-step approach to determine its key characteristics: center, eccentricity, vertices, foci, equation of directrix, and lengths of axes. We will delve into the equation $3 x^2-4 y^2-12 x-16 y-94=0$, unraveling its secrets and revealing the hyperbola's essence.
Demystifying the Hyperbola: A Journey into Conic Sections
Before we dive into the specifics of our given equation, let's first establish a solid understanding of the hyperbola itself. A hyperbola is defined as the locus of all points where the difference of the distances from two fixed points, called foci, is constant. This seemingly simple definition gives rise to a rich geometrical structure with several key elements that define its shape and position.
The center of the hyperbola is the midpoint of the segment connecting the two foci. It serves as the hyperbola's central point of symmetry. The vertices are the points where the hyperbola intersects its principal axis, which is the line passing through the foci. The distance between the vertices is the length of the transverse axis, while the distance between the co-vertices (points on the conjugate axis) is the length of the conjugate axis. The eccentricity of a hyperbola, denoted by e, is a measure of how much the hyperbola deviates from a circular shape. It is defined as the ratio of the distance between the foci to the length of the transverse axis. For a hyperbola, e is always greater than 1. The directrices are two lines perpendicular to the principal axis, located at a distance of a/e from the center, where a is the semi-transverse axis. These lines play a crucial role in the hyperbola's definition and properties. Understanding these fundamental concepts is paramount to analyzing and interpreting hyperbolas effectively. By grasping these definitions, we lay the groundwork for a deeper exploration of the hyperbola's characteristics, enabling us to confidently tackle the task of extracting its key parameters from a given equation.
Transforming the Equation: Completing the Square
Our journey to uncover the hyperbola's secrets begins with the given equation: $3 x^2-4 y^2-12 x-16 y-94=0$. This equation, while representing a hyperbola, is not in a readily usable form. To extract the essential parameters, we need to transform it into the standard form of a hyperbola equation. The standard form allows us to easily identify the center, semi-axes, and orientation of the hyperbola. The key technique we'll employ to achieve this transformation is completing the square. This algebraic method allows us to rewrite quadratic expressions in a more convenient form, revealing the underlying structure of the hyperbola equation. Let's embark on this transformation step-by-step.
First, we group the x terms and the y terms together: $(3 x^2-12 x) + (-4 y^2-16 y) - 94 = 0$. Next, we factor out the coefficients of the squared terms: $3(x^2-4 x) - 4(y^2+4 y) - 94 = 0$. Now comes the crucial step: completing the square for both the x and y expressions. To complete the square for an expression of the form $x^2 + bx$, we add and subtract $(b/2)^2$. For the x expression, $x^2 - 4x$, we have $b = -4$, so $(b/2)^2 = (-4/2)^2 = 4$. Similarly, for the y expression, $y^2 + 4y$, we have $b = 4$, so $(b/2)^2 = (4/2)^2 = 4$. Adding and subtracting these values inside the parentheses, we get: $3(x^2-4 x + 4 - 4) - 4(y^2+4 y + 4 - 4) - 94 = 0$. Now we can rewrite the expressions inside the parentheses as perfect squares: $3((x-2)^2 - 4) - 4((y+2)^2 - 4) - 94 = 0$. Distributing the constants, we have: $3(x-2)^2 - 12 - 4(y+2)^2 + 16 - 94 = 0$. Combining the constant terms, we get: $3(x-2)^2 - 4(y+2)^2 - 90 = 0$. Finally, we move the constant term to the right side and divide by 90 to obtain the standard form: $\frac{(x-2)^2}{30} - \frac{(y+2)^2}{\frac{45}{2}} = 1$. This standard form unveils the hyperbola's key features, setting the stage for us to extract its center, eccentricity, vertices, foci, directrix, and axes.
Decoding the Standard Form: Unveiling Hyperbola's Parameters
With the equation now in standard form, $\frac(x-2)^2}{30} - \frac{(y+2)^2}{\frac{45}{2}} = 1$, we can readily identify the hyperbola's key parameters. The standard form equation for a hyperbola with a horizontal transverse axis is given by{a^2} - \frac{(y-k)2}{b2} = 1$, where $(h, k)$ is the center, a is the semi-transverse axis, and b is the semi-conjugate axis. By comparing our transformed equation with the standard form, we can extract the following information:
- Center: The center of the hyperbola is the point $(h, k)$. In our case, $h = 2$ and $k = -2$, so the center is $(2, -2)$. This point serves as the hyperbola's central anchor, around which its two branches extend.
- Semi-transverse axis: The semi-transverse axis, a, is the square root of the denominator under the $(x-h)^2$ term. In our case, $a^2 = 30$, so $a = \sqrt{30}$. This value represents the distance from the center to each vertex along the transverse axis.
- Semi-conjugate axis: The semi-conjugate axis, b, is the square root of the denominator under the $(y-k)^2$ term. In our case, $b^2 = \frac{45}{2}$, so $b = \sqrt{\frac{45}{2}} = \frac{3\sqrt{5}}{\sqrt{2}} = \frac{3\sqrt{10}}{2}$. This value represents the distance from the center to each co-vertex along the conjugate axis.
- Eccentricity: The eccentricity, e, is a measure of how much the hyperbola deviates from a circular shape. It is calculated using the formula $e = \sqrt1 + \frac{b2}{a2}}$. Plugging in our values for a and b, we get{2}}{30}} = \sqrt{1 + \frac{45}{60}} = \sqrt{1 + \frac{3}{4}} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2}$. Since e is greater than 1, it confirms that we are indeed dealing with a hyperbola.
Having deciphered these fundamental parameters, we are now poised to determine the vertices, foci, equation of the directrix, and lengths of the axes, further illuminating the hyperbola's characteristics.
Delving Deeper: Vertices, Foci, Directrix, and Axes
Building upon the parameters we've extracted, let's now determine the vertices, foci, equation of the directrix, and lengths of the axes, completing our comprehensive analysis of the hyperbola. These elements provide a more detailed understanding of the hyperbola's shape, position, and orientation.
- Vertices: The vertices are the points where the hyperbola intersects its transverse axis. For a hyperbola with a horizontal transverse axis, the vertices are located at $(h \pm a, k)$. Using our values $h = 2$, $k = -2$, and $a = \sqrt30}$, the vertices are, -2)$ and $(2 - \sqrt{30}, -2)$. These points mark the endpoints of the hyperbola's main axis.
- Foci: The foci are the two fixed points that define the hyperbola. They are located at a distance of c from the center along the transverse axis, where $c = \sqrta^2 + b^2}$. In our case, $c = \sqrt{30 + \frac{45}{2}} = \sqrt{\frac{105}{2}} = \sqrt{\frac{210}{4}} = \frac{\sqrt{210}}{2}$. The foci are located at $(h \pm c, k)$, so they are}{2}, -2)$ and $(2 - \frac{\sqrt{210}}{2}, -2)$. These points are crucial in understanding the hyperbola's reflective properties.
- Equation of Directrix: The directrices are two lines perpendicular to the transverse axis, located at a distance of a/e from the center. The equations of the directrices are given by $x = h \pm \fraca}{e}$. Using our values, we havee} = \frac{\sqrt{30}}{\frac{\sqrt{7}}{2}} = \frac{2\sqrt{30}}{\sqrt{7}} = \frac{2\sqrt{210}}{7}$. Therefore, the equations of the directrices are}{7}$ and $x = 2 - \frac{2\sqrt{210}}{7}$. These lines play a significant role in the hyperbola's geometric definition.
- Length of Axes: The length of the transverse axis is $2a = 2\sqrt{30}$, and the length of the conjugate axis is $2b = 2 * \frac{3\sqrt{10}}{2} = 3\sqrt{10}$. These lengths provide a quantitative measure of the hyperbola's dimensions.
By meticulously calculating these elements, we have painted a complete picture of the hyperbola defined by the equation $3 x^2-4 y^2-12 x-16 y-94=0$. Our journey has taken us from the initial equation to a comprehensive understanding of its geometrical properties.
Conclusion: A Triumph in Hyperbola Unveiling
In this exploration, we embarked on a journey to unravel the characteristics of a hyperbola defined by the equation $3 x^2-4 y^2-12 x-16 y-94=0$. Through the methodical process of completing the square, we transformed the equation into its standard form, which served as a key to unlocking the hyperbola's secrets. We successfully identified the center at $(2, -2)$, the eccentricity as $\frac{\sqrt{7}}{2}$, the vertices at $(2 + \sqrt{30}, -2)$ and $(2 - \sqrt{30}, -2)$, and the foci at $(2 + \frac{\sqrt{210}}{2}, -2)$ and $(2 - \frac{\sqrt{210}}{2}, -2)$. We also determined the equations of the directrices as $x = 2 + \frac{2\sqrt{210}}{7}$ and $x = 2 - \frac{2\sqrt{210}}{7}$, and the lengths of the axes as $2\sqrt{30}$ for the transverse axis and $3\sqrt{10}$ for the conjugate axis. This comprehensive analysis showcases the power of algebraic manipulation and the beauty of conic sections. By understanding the underlying principles and applying the appropriate techniques, we can confidently dissect and interpret any hyperbola equation, revealing its essential geometrical properties. This knowledge empowers us to utilize hyperbolas in various applications, from modeling physical phenomena to designing engineering structures. The hyperbola, once a mysterious curve, now stands revealed, its secrets unveiled through the lens of mathematical analysis.
