Finding Hyperbola Equation From Asymptotes And Vertex
Hey there, math enthusiasts! Today, we're diving into the fascinating world of hyperbolas. Specifically, we're going to tackle the challenge of finding the equation of a hyperbola when we're given its asymptotes and one vertex. It might sound a bit intimidating at first, but trust me, we'll break it down step by step, and you'll be a hyperbola-solving pro in no time!
Understanding Hyperbolas: A Quick Refresher
Before we jump into the problem, let's quickly revisit what hyperbolas are all about. A hyperbola is a type of conic section, which basically means it's a curve formed when you slice a cone with a plane at a certain angle. Unlike an ellipse, which is a closed curve, a hyperbola has two separate branches that open away from each other. Think of it as two parabolas facing opposite directions.
Key features of a hyperbola include:
- Center: The midpoint between the two vertices.
- Vertices: The points where the hyperbola intersects its main axis (the transverse axis).
- Foci: Two points inside the hyperbola that are used in its definition.
- Asymptotes: These are the lines that the hyperbola approaches as it extends infinitely. They act as guides for the shape of the hyperbola.
The standard form equation of a hyperbola centered at the origin depends on whether it opens horizontally or vertically:
- Horizontal Hyperbola: (x^2 / a^2) - (y^2 / b^2) = 1
- Vertical Hyperbola: (y^2 / a^2) - (x^2 / b^2) = 1
Where:
- 'a' is the distance from the center to each vertex.
- 'b' is related to the distance between the branches of the hyperbola.
The asymptotes of a hyperbola centered at the origin have the equations:
- y = (b/a)x and y = -(b/a)x (for a horizontal hyperbola)
- y = (a/b)x and y = -(a/b)x (for a vertical hyperbola)
With these basics in mind, we're well-equipped to tackle our problem!
Problem Statement: Finding the Equation
Okay, let's get down to business. Our mission, should we choose to accept it (and we do!), is to find the equation of a hyperbola that satisfies the following conditions:
- Asymptotes: y = (1/4)x and y = -(1/4)x
- One vertex: (12, 0)
Let's break down our strategy:
- Determine the Orientation: The asymptotes will give us a clue whether the hyperbola opens horizontally or vertically.
- Find 'a': The vertex will help us determine the value of 'a', the distance from the center to the vertices.
- Relate Asymptotes to 'a' and 'b': We'll use the equations of the asymptotes to find the relationship between 'a' and 'b'.
- Solve for 'b': Using the information from steps 2 and 3, we'll solve for 'b'.
- Write the Equation: Finally, we'll plug the values of 'a' and 'b' into the appropriate standard form equation.
Step-by-Step Solution
Let's get started, guys!
Step 1: Determine the Orientation
Notice that the given asymptotes are y = (1/4)x and y = -(1/4)x. These are in the form y = ±(some constant)x. This tells us that the hyperbola is centered at the origin (0, 0). Also, since the vertex is given as (12, 0), which lies on the x-axis, we know that the hyperbola opens horizontally. This is a crucial piece of information! We now know our equation will look like (x^2 / a^2) - (y^2 / b^2) = 1.
Step 2: Find 'a'
We're given that one vertex is (12, 0). Since the hyperbola is centered at the origin and opens horizontally, the vertices are located at (±a, 0). Therefore, we can directly see that a = 12. Awesome! We've found 'a'. This value represents the distance from the center of the hyperbola to each of its vertices along the transverse axis.
Step 3: Relate Asymptotes to 'a' and 'b'
For a horizontal hyperbola, the asymptotes have the equations y = ±(b/a)x. We're given the asymptotes as y = (1/4)x and y = -(1/4)x. Comparing these, we can see that b/a = 1/4. This equation establishes a direct relationship between the parameters 'a' and 'b', which we will use to solve for 'b' in the next step. Understanding this relationship is key to linking the geometry of the hyperbola to its algebraic representation.
Step 4: Solve for 'b'
We know that a = 12 and b/a = 1/4. We can now substitute the value of 'a' into the equation b/a = 1/4 to solve for 'b'.
b / 12 = 1/4
Multiply both sides by 12:
b = (1/4) * 12
b = 3
Fantastic! We've found 'b'. This value is related to the conjugate axis of the hyperbola and influences the steepness of its branches.
Step 5: Write the Equation
We have all the pieces we need! We know the hyperbola opens horizontally, a = 12, and b = 3. Let's plug these values into the standard form equation for a horizontal hyperbola:
(x^2 / a^2) - (y^2 / b^2) = 1
(x^2 / 12^2) - (y^2 / 3^2) = 1
(x^2 / 144) - (y^2 / 9) = 1
And there you have it! That's the equation of our hyperbola! We have successfully translated the given geometric conditions into a precise algebraic equation, which fully describes the hyperbola's shape and position in the coordinate plane.
Final Answer
The equation of the hyperbola is (x^2 / 144) - (y^2 / 9) = 1.
Key Takeaways
- Understanding the standard form equations of hyperbolas is crucial.
- The asymptotes provide valuable information about the hyperbola's orientation and the relationship between 'a' and 'b'.
- The vertex helps determine the value of 'a'.
- By carefully combining these pieces of information, we can successfully find the equation of the hyperbola.
Practice Makes Perfect
Guys, the best way to master hyperbola problems is to practice! Try working through similar examples with different given conditions. You'll soon become a hyperbola expert!
I hope this explanation was helpful. Keep exploring the fascinating world of mathematics!