Hyperbola Foci: Find Coordinates & Distance | Math Guide
Hey guys! Today, we're diving deep into the fascinating world of hyperbolas, specifically focusing on how to find the coordinates of the foci and the distance between them. We'll be tackling a common problem using the general equation of a hyperbola. So, let's get started and break down the solution step-by-step!
Understanding Hyperbolas and Their General Equation
First things first, let's make sure we're all on the same page about what a hyperbola actually is. A hyperbola is a type of conic section, which basically means it's a curve formed when a plane intersects a double cone. Unlike an ellipse, which is like a stretched circle, a hyperbola has two separate branches that open away from each other. The general equation of a hyperbola centered at the origin is given by:
(x²/a²) - (y²/b²) = 1
Where:
- 'a' is the distance from the center to each vertex along the transverse axis (the axis that passes through the foci and vertices).
- 'b' is the distance from the center to each vertex along the conjugate axis (the axis perpendicular to the transverse axis).
Another key element of a hyperbola are its foci. These are two fixed points inside the hyperbola such that for any point on the hyperbola, the difference in distances to the two foci is constant. This is a crucial property that helps define the shape of the hyperbola. The distance from the center to each focus is denoted by 'c', and it's related to 'a' and 'b' by the following equation:
c² = a² + b²
This relationship is super important because it allows us to calculate the distance to the foci once we know 'a' and 'b'. The foci always lie on the transverse axis, symmetrically placed about the center. For a hyperbola with a horizontal transverse axis (like the one we'll be dealing with in this problem), the foci are located at coordinates (±c, 0). Understanding these fundamental concepts about hyperbolas is key to solving problems related to their foci and other properties. So, keep these definitions and equations in mind as we move on to the specific problem at hand. Now that we have a solid foundation, let's get back to our original equation and see how we can apply these principles to find the coordinates of the foci.
Solving for the Foci of the Given Hyperbola
Okay, now let's dive into the specific problem we were given. The general equation of the hyperbola is:
4x² - 9y² - 36 = 0
The first step in finding the coordinates of the foci is to rewrite this equation in the standard form that we discussed earlier, which is (x²/a²) - (y²/b²) = 1. To do this, we need to get the constant term (-36) to the right side of the equation and then divide everything by that constant. So, let's start by adding 36 to both sides:
4x² - 9y² = 36
Next, we divide both sides of the equation by 36 to get 1 on the right side:
(4x²/36) - (9y²/36) = 1
Now, let's simplify the fractions:
(x²/9) - (y²/4) = 1
Aha! We've now got our equation in the standard form. From this, we can easily identify the values of a² and b²:
- a² = 9, so a = 3
- b² = 4, so b = 2
Remember that 'a' is the distance from the center to the vertices along the transverse axis, and 'b' is related to the conjugate axis. Now, to find the foci, we need to calculate 'c', which is the distance from the center to each focus. We use the relationship c² = a² + b²:
c² = 9 + 4 = 13
Taking the square root of both sides, we get:
c = √13
Since our hyperbola is in the form (x²/a²) - (y²/b²) = 1, the transverse axis is horizontal, and the foci are located at (±c, 0). Therefore, the coordinates of the foci are (±√13, 0). But wait, let's look at the answer choices. None of them have √13! This means we might have made a small mistake or misinterpreted something. Let's backtrack and make sure everything's correct. (After reviewing the steps, it seems there was no mistake in the calculations. The value of c is indeed √13, which is not directly present in the options.)
Correcting and Verifying the Solution Process
Alright, guys, let's take a step back and double-check our work. Sometimes a fresh look can help us spot any tiny errors or misinterpretations. We started with the equation 4x² - 9y² - 36 = 0, and we meticulously transformed it into the standard form (x²/9) - (y²/4) = 1. We correctly identified a² as 9 and b² as 4, which gave us a = 3 and b = 2. Then we used the fundamental relationship c² = a² + b² to find c, the distance from the center to each focus.
We calculated c² = 9 + 4 = 13, and consequently, c = √13. The foci of a hyperbola in this standard form are indeed located at (±c, 0), which led us to (±√13, 0). Now, looking at the answer choices provided, we have:
a) (±3, 0) b) (±6, 0) c) (±4, 0) d) (±5, 0)
It's clear that (±√13, 0) doesn't directly match any of these options. However, this is a typical situation where we need to carefully consider what the question is asking and how the answer choices are presented. The value √13 is approximately 3.61. This means the foci are located approximately at (±3.61, 0). Given the options, we need to choose the one that is closest to our calculated value. The closest integer value to 3.61 is 4, but that's not one of our direct results.
Let's think about this logically. If the question were slightly different, or if the options were approximations, we'd choose the closest one. But in this precise scenario, it appears there might be a subtle twist or perhaps an error in the provided choices. It's crucial to understand that our process—converting the equation, finding a and b, calculating c, and understanding the foci coordinates—is mathematically sound. We've followed all the correct steps.
Given the discrepancy between our result (±√13, 0) and the provided options, it's essential to highlight the importance of double-checking not just our work, but also the question and the options themselves. Sometimes, errors occur in the transcription of problems or in the answer keys. In this case, while our calculated answer doesn't perfectly align with the options, we are confident in our methodology and the correctness of our calculations. The most accurate answer based on our work is (±√13, 0), which isn't directly available, suggesting a potential issue with the choices provided.
Final Thoughts and the Correct Answer
Alright, guys, after a thorough review and verification of our solution process, we've arrived at a crucial point. We've confidently determined that the coordinates of the foci for the given hyperbola equation 4x² - 9y² - 36 = 0 are (±√13, 0). This result stems from correctly converting the equation to its standard form, identifying the values of a and b, and then using the relationship c² = a² + b² to find c, the distance from the center to each focus.
However, as we noted earlier, our calculated coordinates (±√13, 0) do not directly match any of the provided answer choices:
a) (±3, 0) b) (±6, 0) c) (±4, 0) d) (±5, 0)
This discrepancy highlights an important aspect of problem-solving in mathematics: sometimes, the correct answer might not be explicitly listed among the options. This could be due to a variety of reasons, such as a mistake in the question itself, an error in the answer choices, or the need for a deeper level of approximation or interpretation.
In this specific scenario, while we are certain of our calculations, we must make a choice based on the given alternatives. Since √13 is approximately 3.61, the closest value among the options for the x-coordinate of the foci is 4. However, we must recognize that (±4, 0) is not the exact answer, but the closest one available.
Therefore, based on the choices provided and the understanding that we are selecting the nearest option, the most suitable answer is:
c) (±4, 0)
It's crucial to emphasize that this choice is made recognizing the limitations of the options and the exactness of our calculated result. In a real-world scenario, if the precise answer (±√13, 0) was required, it would be essential to communicate the discrepancy and seek clarification.
So, there you have it! We've successfully navigated through finding the foci of a hyperbola, highlighting the importance of precise calculations and thoughtful interpretation of results. Remember, guys, math is not just about arriving at an answer but also understanding the journey and the nuances along the way. Keep practicing, keep questioning, and you'll become math whizzes in no time!