Hyperbola Equation: Finding The Missing Value

Let's dive into the fascinating world of hyperbolas and tackle the challenge of finding a missing value in its equation. This article will walk you through the steps to solve this problem, ensuring you understand the underlying concepts along the way. We'll explore the properties of hyperbolas, their equations, and how the directrix plays a crucial role in determining the shape and characteristics of these conic sections. So, buckle up and get ready to unravel the mystery of the missing value!

Understanding Hyperbolas and Their Equations

In order to find the missing positive value in the given equation, let's first establish a solid understanding of what a hyperbola is and how its equation is structured. A hyperbola is a type of conic section, which means it's a curve formed by the intersection of a plane and a double cone. Think of it as two mirrored parabolas opening away from each other. The general form of a hyperbola centered at the origin, which is relevant to our problem, is:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

In this equation:

• x and y are the coordinates of any point on the hyperbola.
• a is the distance from the center of the hyperbola to each vertex along the transverse axis (the axis that passes through the foci).
• b is related to the distance from the center to the co-vertices along the conjugate axis (the axis perpendicular to the transverse axis).

Our given equation is:

$$\frac{x^2}{24^2} - \frac{y^2}{[\ldots]^2} = 1$$

We can see that a = 24. Our task is to determine the value of b, the missing positive value, using the information provided about the directrix. The directrix is a line associated with the hyperbola, and its position is related to the hyperbola's shape and dimensions. Understanding this relationship is key to solving our problem. We'll delve into the directrix in the next section.

The Directrix and Its Role

The directrix is a crucial element in defining a hyperbola. A hyperbola has two directrices, both perpendicular to the transverse axis. The defining property of a hyperbola is that for any point on the hyperbola, the ratio of its distance to a focus (a fixed point) and its distance to the corresponding directrix is a constant greater than 1. This constant is called the eccentricity, denoted by e. The eccentricity helps us understand how “stretched” the hyperbola is. A higher eccentricity means a wider hyperbola. For a hyperbola centered at the origin with the equation we discussed earlier, the equations of the directrices are:

$$x = \pm \frac{a}{e}$$

Where:

• a is the distance from the center to the vertex, as we defined earlier.
• e is the eccentricity of the hyperbola.

We are given that one directrix of our hyperbola is:

$$x = \frac{576}{26}$$

Since our hyperbola is centered at the origin and opens along the x-axis (because the $x^2$ term is positive), this directrix is to the right of the center. We also know that a = 24. Therefore, we can set up the following equation:

$$\frac{576}{26} = \frac{24}{e}$$

Solving for e, we get:

$$e = \frac{24}{\frac{576}{26}} = \frac{24 \cdot 26}{576} = \frac{26}{24} = \frac{13}{12}$$

So, the eccentricity of our hyperbola is $\frac{13}{12}$. This is a critical piece of information. Now that we know the eccentricity and the value of a, we can move on to finding the value of b, which is the missing piece in our equation. This involves understanding the relationship between a, b, and e, which we'll explore in the next section.

Connecting a, b, and e

Now, let's explore the fundamental relationship between the parameters a, b, and e in a hyperbola. This connection is key to finding the missing value in our equation. The relationship is given by the equation:

$$e^2 = 1 + \frac{b^2}{a^2}$$

This equation tells us how the eccentricity e is related to the distances a and b, which define the shape of the hyperbola. We already know:

• a = 24
• e = \frac{13}{12}

We can now substitute these values into the equation and solve for b:

$$\left(\frac{13}{12}\right)^2 = 1 + \frac{b^2}{24^2}$$

$$\frac{169}{144} = 1 + \frac{b^2}{576}$$

Now, we need to isolate the term with b^2. Subtract 1 from both sides:

$$\frac{169}{144} - 1 = \frac{b^2}{576}$$

$$\frac{169 - 144}{144} = \frac{b^2}{576}$$

$$\frac{25}{144} = \frac{b^2}{576}$$

Next, multiply both sides by 576:

$$b^2 = \frac{25}{144} \cdot 576$$

$$b^2 = 25 \cdot 4$$

$$b^2 = 100$$

Finally, take the square root of both sides to find b. Since we're looking for the positive value, we have:

$$b = \sqrt{100} = 10$$

Therefore, the missing positive value in the equation is 10. We have successfully navigated through the properties of hyperbolas, understood the role of the directrix, and utilized the relationship between a, b, and e to arrive at the solution. In the next section, we'll summarize our findings and present the complete equation of the hyperbola.

The Complete Hyperbola Equation

Now that we've found the missing value, let's put it all together and write out the complete equation of the hyperbola. We started with the equation:

$$\frac{x^2}{24^2} - \frac{y^2}{[\ldots]^2} = 1$$

We determined that the missing value is b = 10. Therefore, the complete equation of the hyperbola is:

$$\frac{x^2}{24^2} - \frac{y^2}{10^2} = 1$$

$$\frac{x^2}{576} - \frac{y^2}{100} = 1$$

This equation represents a hyperbola centered at the origin, opening along the x-axis, with a distance of 24 from the center to each vertex and a value of 10 for b, which is related to the distance to the co-vertices. We also know that this hyperbola has a directrix at $x = \frac{576}{26}$, which we used to find the eccentricity and ultimately the value of b. We have successfully solved the problem and now have a complete understanding of this hyperbola and its defining equation.

In conclusion, by understanding the properties of hyperbolas, the significance of the directrix, and the relationships between the parameters a, b, and e, we were able to successfully find the missing positive value in the equation. This exercise highlights the interconnectedness of these concepts in conic sections and demonstrates how we can use them to solve geometric problems. For further exploration of hyperbolas and other conic sections, you can visit resources like Khan Academy's Conic Sections. This trusted website offers comprehensive explanations, examples, and practice problems to deepen your understanding of these fascinating mathematical concepts.