Finding Foci Of Hyperbola: A Step-by-Step Guide

Alright guys, let's dive into the fascinating world of hyperbolas! Specifically, we're going to tackle the question: Given the general equation of a hyperbola 4x² - y² - 8x + 4y - 12 = 0, how do we actually find its foci? It might seem daunting at first, but trust me, by breaking it down into manageable steps, we can conquer this mathematical beast. The main goal here is to transform the given general equation into the standard form of a hyperbola's equation. Once we have that, identifying the foci becomes a whole lot easier. So, buckle up, and let's get started!

1. Rearranging and Completing the Square

First things first, let's gather like terms and rearrange the equation a bit. We want to group the x terms together and the y terms together. This will make it easier to complete the square, which is a crucial step in getting to the standard form. So, we start with: 4x² - y² - 8x + 4y - 12 = 0. Now, let's rearrange it to look like this: (4x² - 8x) - (y² - 4y) = 12. See how we've grouped the x and y terms? Don't forget to move the constant term to the right side of the equation.

Now comes the fun part: completing the square. Remember, the goal here is to turn those expressions in parentheses into perfect square trinomials. For the x terms, we have 4x² - 8x. Factor out the coefficient of the x² term, which is 4: 4(x² - 2x). To complete the square inside the parentheses, we need to add and subtract (2/2)² = 1. So, we get 4(x² - 2x + 1 - 1) = 4((x - 1)² - 1) = 4(x - 1)² - 4. For the y terms, we have y² - 4y. Here, the coefficient of the y² term is already 1, so we just need to add and subtract (4/2)² = 4. This gives us (y² - 4y + 4 - 4) = (y - 2)² - 4. Therefore, -(y² - 4y) = -((y - 2)² - 4) = -(y - 2)² + 4.

2. Standard Form Transformation

Now that we've completed the square, let's plug those expressions back into our equation. We had (4x² - 8x) - (y² - 4y) = 12. Substituting our completed square expressions, we get 4(x - 1)² - 4 - (y - 2)² + 4 = 12. Simplify this to 4(x - 1)² - (y - 2)² = 12. To get this into the standard form of a hyperbola, we need to divide both sides by 12: [4(x - 1)²]/12 - [(y - 2)²]/12 = 1. This simplifies to [(x - 1)²]/3 - [(y - 2)²]/12 = 1. Aha! We're getting closer to the standard form of a hyperbola, which is either [(x - h)²]/a² - [(y - k)²]/b² = 1 (for a horizontal hyperbola) or [(y - k)²]/a² - [(x - h)²]/b² = 1 (for a vertical hyperbola). In our case, it's a horizontal hyperbola. Comparing our equation [(x - 1)²]/3 - [(y - 2)²]/12 = 1 to the standard form, we can identify the center, a², and b². The center of the hyperbola is (h, k) = (1, 2). We also have a² = 3 and b² = 12. This means a = √3 and b = √12 = 2√3. Remember, the standard form is super important because it gives us all the key information about the hyperbola.

3. Finding the Foci

Okay, so we've got the standard form, the center, a, and b. Now, the moment we've all been waiting for: finding the foci. The distance from the center to each focus is denoted by c, and it's related to a and b by the equation c² = a² + b². In our case, c² = 3 + 12 = 15. Therefore, c = √15. Since our hyperbola is horizontal (because the x² term comes first), the foci will be located at (h ± c, k). We know that (h, k) = (1, 2) and c = √15. So, the coordinates of the foci are (1 + √15, 2) and (1 - √15, 2). And there you have it! We've successfully found the foci of the hyperbola.

Summary of the steps

Let's recap the steps we took to find the foci of the hyperbola:

1. Rearrange and Complete the Square: Group x and y terms, and complete the square for both.
2. Standard Form Transformation: Rewrite the equation in the standard form of a hyperbola: [(x - h)²]/a² - [(y - k)²]/b² = 1 or [(y - k)²]/a² - [(x - h)²]/b² = 1.
3. Finding the Foci: Calculate c using c² = a² + b², and find the foci coordinates using (h ± c, k) for a horizontal hyperbola or (h, k ± c) for a vertical hyperbola.

Key Concepts Recap:

• Hyperbola: A conic section formed by the intersection of a double cone with a plane.
• Foci: Two fixed points inside the hyperbola used in the formal definition of the curve.
• Standard Form: The simplified equation form that reveals key parameters such as center, a, and b.
• Completing the Square: A technique to convert quadratic expressions into perfect square trinomials.

Understanding Hyperbolas: A Deeper Dive

Hyperbolas are not just abstract mathematical concepts; they pop up in various real-world scenarios. For example, the trajectory of a comet as it slingshots around the sun often resembles a hyperbola. The cooling towers of nuclear power plants are hyperbolic structures, chosen for their structural integrity and efficiency. Understanding hyperbolas can also be crucial in fields like navigation and astronomy.

The key parameters of a hyperbola are the center (h, k), the values a and b, and the distance to the foci c. The orientation of the hyperbola (horizontal or vertical) depends on whether the x² term or the y² term is positive in the standard form equation. If the x² term is positive, it's a horizontal hyperbola; if the y² term is positive, it's a vertical hyperbola. Remember that the relationship c² = a² + b² is fundamental in finding the foci.

Completing the square might seem like a tedious process, but it's a powerful tool for dealing with quadratic equations and conic sections. By completing the square, we can transform general equations into standard forms, which makes it much easier to identify key parameters and solve problems. Practice this technique, and it will become second nature!

Common Mistakes to Avoid

When working with hyperbolas, there are a few common mistakes that students often make. One common mistake is messing up the signs when completing the square, especially when there's a negative sign in front of the y² term. Always double-check your work to make sure you've handled the signs correctly.

Another common mistake is confusing the values of a and b. Remember that a² is always associated with the positive term in the standard form equation. If the x² term is positive, then a² is under the (x - h)² term; if the y² term is positive, then a² is under the (y - k)² term. Also, be careful when calculating c using c² = a² + b². Make sure you add a² and b², not subtract them.

Finally, don't forget to consider the orientation of the hyperbola when finding the foci. If it's a horizontal hyperbola, the foci are located at (h ± c, k); if it's a vertical hyperbola, the foci are located at (h, k ± c). Knowing the orientation is crucial for getting the correct coordinates.

Practice Problems

To solidify your understanding of finding the foci of a hyperbola, try working through some practice problems. Here's one to get you started:

Problem: Find the foci of the hyperbola given by the equation 9x² - 4y² - 18x + 16y - 43 = 0.

Solution:

1. Rearrange and Complete the Square: (9x² - 18x) - (4y² - 16y) = 43 9(x² - 2x) - 4(y² - 4y) = 43 9(x² - 2x + 1 - 1) - 4(y² - 4y + 4 - 4) = 43 9((x - 1)² - 1) - 4((y - 2)² - 4) = 43 9(x - 1)² - 9 - 4(y - 2)² + 16 = 43 9(x - 1)² - 4(y - 2)² = 36
2. Standard Form Transformation: [9(x - 1)²]/36 - [4(y - 2)²]/36 = 1 [(x - 1)²]/4 - [(y - 2)²]/9 = 1
3. Finding the Foci: a² = 4, b² = 9, c² = a² + b² = 4 + 9 = 13, c = √13 Center: (1, 2) Foci: (1 ± √13, 2)

So, the foci of the hyperbola are (1 + √13, 2) and (1 - √13, 2).

Conclusion

Finding the foci of a hyperbola might seem challenging at first, but by following these steps and practicing regularly, you'll become a pro in no time! Remember to rearrange the equation, complete the square, transform it into standard form, and then use the relationship c² = a² + b² to find the distance from the center to the foci. Good luck, and happy calculating!

So there you have it. Armed with this knowledge, you're well-equipped to tackle any hyperbola equation that comes your way. Keep practicing, and soon you'll be spotting those foci like a mathematical eagle! Remember, the key is to break down the problem into smaller, manageable steps, and don't be afraid to ask for help when you need it. Now go forth and conquer those hyperbolas!