Hyperbola Equation: Vertices & Foci Explained

Hey math enthusiasts! Today, we're diving deep into the fascinating world of hyperbolas. Specifically, we're going to tackle a problem where we need to find the equation of a hyperbola given its vertices and foci. It might sound intimidating at first, but trust me, we'll break it down step-by-step so it's super clear. Think of it like this: we're mathematical detectives, and the equation of the hyperbola is the mystery we're solving. So, grab your thinking caps, and let's get started!

Understanding Hyperbolas: A Quick Refresher

Before we jump into the problem, let's do a quick recap on what hyperbolas are all about. Hyperbolas are conic sections, which basically means they're the curves you get when you slice a double cone with a plane at a certain angle. Imagine two identical curves facing away from each other – that's essentially a hyperbola. Now, these curves have some key features we need to know about.

• Vertices: These are the points where the hyperbola curves most sharply. Think of them as the “corners” of the hyperbola.
• Foci: These are two special points inside the hyperbola that define its shape. The distance between any point on the hyperbola and the two foci has a constant difference.
• Center: This is the midpoint between the vertices (and also the midpoint between the foci). It's like the central hub of the hyperbola.
• Major Axis: This is the line segment connecting the vertices. It's the longer axis of the hyperbola.
• Minor Axis: This is the line segment perpendicular to the major axis, passing through the center. It's the shorter axis of the hyperbola.
• Asymptotes: These are lines that the hyperbola approaches as it extends infinitely. They act like guidelines for the hyperbola's shape.

Understanding these key features is crucial for finding the equation of a hyperbola. It's like knowing the pieces of a puzzle before you start putting it together. Now that we've refreshed our memory, let's get back to our specific problem.

The Challenge: Finding the Hyperbola's Equation

Here's the problem we're tackling: "The vertices of a hyperbola are located at $(0,-4)$ and $(0,12)$. The foci of the same hyperbola are located at $(0,-6)$ and $(0,14)$. What is the equation of the hyperbola?"

Okay, so we're given the coordinates of the vertices and foci. Our mission, should we choose to accept it (and we do!), is to find the equation of the hyperbola. To do this, we'll use the information we have and the standard form of a hyperbola equation. This is where the detective work really begins!

Step-by-Step Solution: Cracking the Code

Let's break down the solution into manageable steps. We'll follow a logical path, using the given information to piece together the equation. It's like following a treasure map, with each step leading us closer to the final answer.

1. Finding the Center

The center of the hyperbola is the midpoint of the vertices (or the foci). Remember the midpoint formula? It's simply the average of the x-coordinates and the average of the y-coordinates. So, let's find the center:

• Vertices: $(0, -4)$ and $(0, 12)$
• Center's y-coordinate: $(-4 + 12) / 2 = 8 / 2 = 4$
• Since both vertices have an x-coordinate of 0, the center's x-coordinate is also 0.

Therefore, the center of the hyperbola is $(0, 4)$. This is our first major clue! Knowing the center is like finding the base camp before climbing a mountain.

2. Determining the Orientation

Next, we need to figure out whether the hyperbola opens vertically or horizontally. This is crucial because it determines the form of the equation we'll use. Notice that both the vertices and foci have the same x-coordinate (0). This means they lie on a vertical line, the y-axis. Therefore, the hyperbola opens vertically. Think of it like a tall, stretched-out oval.

3. Finding the Value of 'a'

The distance between the center and a vertex is denoted by 'a'. This value is a key parameter in the hyperbola's equation. We already know the center $(0, 4)$ and one of the vertices $(0, -4)$. Let's calculate the distance:

• Distance = |4 - (-4)| = |8| = 8

So, $a = 8$. We've found another important piece of the puzzle!

4. Finding the Value of 'c'

The distance between the center and a focus is denoted by 'c'. This value is also essential for determining the hyperbola's equation. We know the center $(0, 4)$ and one of the foci $(0, -6)$. Let's calculate the distance:

• Distance = |4 - (-6)| = |10| = 10

So, $c = 10$. We're on a roll! We've now found 'a' and 'c', which are like two of the three ingredients in a recipe.

5. Finding the Value of 'b'

Here's where the relationship between 'a', 'b', and 'c' comes into play. For a hyperbola, the relationship is given by the equation: $c^2 = a^2 + b^2$. We know 'a' and 'c', so we can solve for 'b':

• $10^2 = 8^2 + b^2$
• $100 = 64 + b^2$
• $b^2 = 100 - 64 = 36$
• $b = \sqrt{36} = 6$

Great! We've found 'b'. Now we have all the pieces we need to write the equation.

6. Writing the Equation

Since the hyperbola opens vertically, the standard form of the equation is:

rac{(y - k)^2}{a^2} - rac{(x - h)^2}{b^2} = 1

Where $(h, k)$ is the center of the hyperbola. We know:

• $(h, k) = (0, 4)$
• $a = 8$
• $b = 6$

Plugging these values into the equation, we get:

rac{(y - 4)^2}{8^2} - rac{(x - 0)^2}{6^2} = 1

Simplifying, we have:

rac{(y - 4)^2}{64} - rac{x^2}{36} = 1

The Solution: We Cracked the Code!

Therefore, the equation of the hyperbola is $\frac{(y - 4)^2}{64} - \frac{x^2}{36} = 1$. Woohoo! We successfully found the equation by carefully analyzing the given information and applying the properties of hyperbolas. It's like we just completed a challenging math quest and found the hidden treasure.

Key Takeaways: Mastering Hyperbolas

Let's recap what we've learned in this adventure:

• Understand the key features of a hyperbola: vertices, foci, center, major axis, minor axis, and asymptotes.
• Find the center: It's the midpoint of the vertices (or foci).
• Determine the orientation: Does it open vertically or horizontally?
• Find 'a', 'b', and 'c': These values are crucial for the equation.
• Use the relationship $c^2 = a^2 + b^2$ to find the missing value.
• Write the equation using the standard form, plugging in the values you found.

By following these steps, you can confidently tackle any hyperbola equation problem. Remember, practice makes perfect! The more you work with hyperbolas, the more comfortable you'll become with their properties and equations. So, keep exploring, keep learning, and keep having fun with math!

I hope this breakdown was helpful and made the concept of finding the equation of a hyperbola much clearer for you. Until next time, happy problem-solving, guys! You've got this! Now go forth and conquer those hyperbolas!