Finding The Value Of B In A Hyperbola Equation

In the realm of conic sections, hyperbolas stand out as fascinating curves with unique properties. Understanding hyperbolas is crucial in various fields, from physics and engineering to astronomy and mathematics. This article delves into a specific problem involving a hyperbola equation and guides you through the process of finding the value of a key parameter. Our focus will be on the equation $\frac{x^2}{24^2} - \frac{y^2}{b^2} = 1$, which represents a hyperbola centered at the origin. We are given that the directrix of this hyperbola is $x = \frac{576}{26}$, and our mission is to determine the value of b. This exploration will not only provide a solution to this particular problem but also enhance your overall understanding of hyperbolas and their properties. To fully grasp the concepts, we will revisit the fundamental characteristics of hyperbolas, such as their standard equation, foci, vertices, and directrices. By connecting these elements, we will develop a clear pathway to solve for b and appreciate the elegance of conic sections. Hyperbolas are defined as the locus of points where the difference of the distances to two fixed points (foci) is constant. This definition gives rise to their distinctive two-branched shape. The standard form of a hyperbola centered at the origin depends on whether the hyperbola opens horizontally or vertically. In our case, the given equation indicates a hyperbola that opens horizontally, making the x-axis the transverse axis and the y-axis the conjugate axis. The values in the denominators of the equation dictate the shape and dimensions of the hyperbola, while the directrix provides additional information about its geometric properties. The directrix is a line associated with the hyperbola that plays a crucial role in its definition and characteristics. By understanding the relationship between the directrix, foci, and vertices, we can unlock the secrets of the hyperbola and solve problems like the one at hand.

Decoding the Hyperbola Equation

To effectively tackle the problem, let's dissect the given hyperbola equation: $\frac{x^2}{24^2} - \frac{y^2}{b^2} = 1$. This equation is in the standard form of a hyperbola centered at the origin, specifically, $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where a and b are constants that define the shape and size of the hyperbola. In our equation, we can immediately identify that $a^2 = 24^2$, which means a = 24. The value of a represents the distance from the center of the hyperbola to its vertices along the transverse axis (the x-axis in this case). The vertices are the points where the hyperbola intersects its transverse axis, and they play a crucial role in visualizing the hyperbola's shape. The parameter b in the equation is related to the conjugate axis of the hyperbola (the y-axis in this case). The value of $b^2$ appears in the denominator of the $y^2$ term, and it influences the hyperbola's overall shape, particularly its curvature. The larger the value of b, the wider the hyperbola opens along the conjugate axis. Our goal is to find the specific value of b that satisfies the given conditions. To do this, we need to connect the equation to the information about the directrix. The directrix of a hyperbola is a line that is related to the foci and vertices. For a hyperbola that opens horizontally, the directrices are vertical lines located at a distance of $\frac{a}{e}$ from the center, where e is the eccentricity of the hyperbola. The eccentricity e is a crucial parameter that describes how “stretched” the hyperbola is. It is defined as $e = \frac{c}{a}$, where c is the distance from the center to each focus. The foci are two fixed points that define the hyperbola, and their location is essential for understanding the hyperbola's geometry. The relationship between a, b, and c is given by the equation $c^2 = a^2 + b^2$. This equation is a cornerstone in understanding the geometry of hyperbolas and will be instrumental in solving our problem. By carefully piecing together these relationships, we can connect the given directrix equation to the hyperbola's parameters and ultimately determine the value of b. The directrix equation, $x = \frac{576}{26}$, provides a critical piece of information that we will use to bridge the gap between the hyperbola's equation and its geometric properties. Understanding these relationships and parameters is key to unlocking the solution and gaining a deeper appreciation for the intricacies of hyperbolas.

Unraveling the Directrix and Eccentricity Connection

The given information states that the directrix of the hyperbola is $x = \frac{576}{26}$. For a hyperbola centered at the origin with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the directrices are vertical lines located at $x = \pm \frac{a}{e}$, where e is the eccentricity of the hyperbola. Since our hyperbola opens horizontally, we focus on the directrix to the right of the center (positive x-value), which corresponds to $x = \frac{a}{e}$. We already know that a = 24 from the equation $\frac{x^2}{24^2} - \frac{y^2}{b^2} = 1$. Therefore, we can set up the equation: $\frac{24}{e} = \frac{576}{26}$. This equation connects the eccentricity e to the given directrix. Solving for e, we multiply both sides by e and by $\frac{26}{576}$ to get: $e = 24 \cdot \frac{26}{576}$. Simplifying the fraction, we find: $e = \frac{24 \cdot 26}{576} = \frac{624}{576} = \frac{13}{12}$. So, the eccentricity of the hyperbola is $\frac{13}{12}$. The eccentricity e is a critical parameter that defines the shape of the hyperbola. Since e > 1 for a hyperbola, this value confirms that we are indeed dealing with a hyperbola and not an ellipse or a parabola. The eccentricity is directly related to the ratio of the distance from the center to the focus (c) and the distance from the center to the vertex (a). Specifically, $e = \frac{c}{a}$. We now have the value of e and a, which allows us to find the distance c from the center to the foci. This distance is essential because it is connected to the value of b, which we are trying to determine. By understanding the relationship between the directrix, eccentricity, and the parameters a, b, and c, we are moving closer to solving for the unknown value of b. The directrix provides a crucial link in this chain of relationships, allowing us to connect the geometric properties of the hyperbola to its algebraic representation. Now that we have found the eccentricity, we can use it to find c, and from there, we can finally solve for b.

Solving for b: The Final Steps

Now that we have determined the eccentricity $e = \frac{13}{12}$ and know that $a = 24$, we can use the relationship $e = \frac{c}{a}$ to find the distance c from the center to the foci. Plugging in the values, we get: $\frac{13}{12} = \frac{c}{24}$. Multiplying both sides by 24, we find: $c = \frac{13}{12} \cdot 24 = 13 \cdot 2 = 26$. So, the distance from the center to each focus is 26 units. The next crucial step is to use the relationship between a, b, and c in a hyperbola, which is given by the equation $c^2 = a^2 + b^2$. We know that $c = 26$ and $a = 24$, so we can plug these values into the equation: $26^2 = 24^2 + b^2$. This equation allows us to directly solve for $b^2$. Calculating the squares, we have: $676 = 576 + b^2$. Subtracting 576 from both sides, we get: $b^2 = 676 - 576 = 100$. Now, we take the square root of both sides to find b: $b = \sqrt{100} = 10$. Therefore, the value of b is 10. This completes our journey to find the value of b in the given hyperbola equation. By carefully analyzing the equation, understanding the properties of hyperbolas, and utilizing the information about the directrix, we were able to systematically solve for the unknown parameter. The process involved connecting the directrix to the eccentricity, finding the distance to the foci, and finally using the relationship between a, b, and c to determine b. This problem showcases the interconnectedness of the various elements of a hyperbola and the power of algebraic manipulation in solving geometric problems. The final answer, $b = 10$, provides a complete description of the hyperbola, allowing us to visualize its shape and position in the coordinate plane. The solution not only answers the specific question but also reinforces the understanding of hyperbolas and their properties. By mastering these concepts, you can confidently tackle similar problems and deepen your appreciation for the beauty and complexity of conic sections.

Conclusion

In conclusion, the value of b in the hyperbola equation $\frac{x^2}{24^2} - \frac{y^2}{b^2} = 1$ with a directrix of $x = \frac{576}{26}$ is 10. Throughout this exploration, we have demonstrated how to connect the equation of a hyperbola to its geometric properties, such as its directrix and eccentricity. By understanding the relationships between the parameters a, b, c, and e, we were able to systematically solve for the unknown value of b. This problem serves as a powerful example of how mathematical concepts intertwine and how a thorough understanding of these concepts can lead to elegant solutions. The journey began with dissecting the hyperbola equation and identifying the known parameter a. We then used the information about the directrix to find the eccentricity e. With e and a in hand, we calculated the distance c from the center to the foci. Finally, we employed the fundamental relationship $c^2 = a^2 + b^2$ to solve for b. This step-by-step approach not only provided the answer but also reinforced the underlying principles of hyperbolas. Mastering these concepts is essential for anyone delving deeper into conic sections and analytic geometry. The ability to connect equations to geometric properties is a valuable skill in mathematics and its applications. This problem highlights the importance of understanding the definitions and relationships associated with hyperbolas, such as the roles of the foci, vertices, directrices, and eccentricity. The solution we presented is not just about finding a numerical answer; it's about developing a deeper understanding of the mathematical structure of hyperbolas. We hope this comprehensive guide has provided clarity and insight into the world of hyperbolas and has empowered you to tackle similar challenges with confidence. Remember, the key to success in mathematics lies in understanding the fundamental concepts and practicing their application. By continuing to explore and connect mathematical ideas, you can unlock the beauty and power of mathematics.

Final Answer: The final answer is (A) 10