Hyperbola Equation Find The Equation Of A Hyperbola In Standard Form

Hey math enthusiasts! Let's dive into the fascinating world of hyperbolas and tackle a problem that might seem tricky at first glance. We're going to figure out the equation of a hyperbola, given its center, focus, and vertex. It's like a mathematical puzzle, and we're here to solve it together, step by step. So, grab your thinking caps, and let's get started!

Understanding Hyperbolas: A Quick Recap

Before we jump into the specifics of our problem, let's refresh our understanding of hyperbolas. A hyperbola is a type of conic section, which basically means it's a curve formed when a plane intersects a double cone. Imagine two cones stacked tip-to-tip, and then slice through them with a plane – the resulting shape is a hyperbola. Now, hyperbolas have some key features that we need to know about:

• Center: This is the midpoint between the two vertices and the two foci of the hyperbola. It's the heart of the hyperbola, so to speak.
• Vertices: These are the points where the hyperbola intersects its main axis of symmetry. Think of them as the 'corners' of the hyperbola.
• Foci: These are two special points inside the hyperbola that play a crucial role in defining its shape. The distance between any point on the hyperbola and the two foci has a constant difference.
• Major Axis: This is the line passing through the center, vertices, and foci. It's the main axis of symmetry of the hyperbola.
• Minor Axis: This is the line passing through the center and perpendicular to the major axis. It helps define the overall shape of the hyperbola.
• Asymptotes: These are the lines that the hyperbola approaches as it extends towards infinity. They act as guidelines for the hyperbola's branches.

The standard form of a hyperbola's equation depends on whether it opens horizontally or vertically. If it opens horizontally, the equation looks like this: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$, where (h, k) is the center. If it opens vertically, the equation is: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$. Remember, the key is to identify the center, the values of 'a' and 'b', and whether the hyperbola opens horizontally or vertically. With that knowledge, we can confidently write the equation.

Problem Statement: Decoding the Hyperbola

Now, let's get back to our specific problem. We're given the following information:

• Center: (7, 0)
• Focus: (7, 5)
• Vertex: (7, 4)

Our mission, should we choose to accept it, is to find the equation of this hyperbola in standard form. It might seem like we don't have much to go on, but trust me, these three points hold the key to unlocking the entire equation. We'll use the properties of hyperbolas and a little bit of algebraic manipulation to crack this mathematical code. So, let's put on our detective hats and start piecing together the puzzle!

Step-by-Step Solution: Cracking the Code

Okay, guys, let's break down how to find the hyperbola's equation step-by-step. We'll use the information we have—the center, focus, and vertex—to figure out the key parameters we need for the standard form equation.

Step 1: Determine the Orientation

The first thing we need to figure out is whether the hyperbola opens vertically or horizontally. To do this, let's look at the coordinates of the center, focus, and vertex. Notice that the x-coordinates of all three points are the same (7). This tells us that the center, focus, and vertex all lie on a vertical line. Therefore, the hyperbola opens vertically. This is a crucial piece of information because it tells us which standard form equation to use.

Step 2: Identify the Center (h, k)

This step is super straightforward because we're already given the center! The center of the hyperbola is (7, 0). So, we know that h = 7 and k = 0. We've already filled in two blanks in our equation template!

Step 3: Find the Value of 'a'

The value of 'a' represents the distance between the center and a vertex. We can calculate this distance using the coordinates of the center (7, 0) and the vertex (7, 4). The distance formula might come to mind, but since the x-coordinates are the same, we can simply find the difference in the y-coordinates:

• a = |4 - 0| = 4

So, a = 4. This is another key piece of the puzzle. Remember, 'a' is always associated with the term that comes first in the standard form equation (in this case, the y-term since it opens vertically).

Step 4: Find the Value of 'c'

The value of 'c' represents the distance between the center and a focus. We can calculate this distance using the coordinates of the center (7, 0) and the focus (7, 5). Again, since the x-coordinates are the same, we just find the difference in the y-coordinates:

• c = |5 - 0| = 5

So, c = 5. This value is important because it helps us find 'b', which we'll do in the next step.

Step 5: Find the Value of 'b'

Now, here's where a special relationship comes into play. For hyperbolas, there's a connection between a, b, and c: $c^2 = a^2 + b^2$. We know a and c, so we can use this equation to solve for b. Let's plug in our values:

• $5^2 = 4^2 + b^2$
• $25 = 16 + b^2$
• $b^2 = 9$
• b = 3

So, b = 3. We've now found all the values we need for our equation!

Step 6: Write the Equation in Standard Form

We've done all the hard work, and now it's time to put it all together. Since the hyperbola opens vertically, we use the standard form equation: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$. Let's plug in our values for h, k, a, and b:

• $\frac{(y-0)^2}{4^2} - \frac{(x-7)^2}{3^2} = 1$

Simplifying, we get:

• $\frac{y^2}{16} - \frac{(x-7)^2}{9} = 1$

And there you have it! We've found the equation of the hyperbola.

Final Equation: The Grand Reveal

The equation of the hyperbola centered at (7,0) with a focus at (7,5) and a vertex at (7,4) is:

$\frac{y^2}{16} - \frac{(x-7)^2}{9} = 1$

Woohoo! We did it! It might have seemed daunting at first, but by breaking it down step by step and using the properties of hyperbolas, we were able to find the equation. High five for conquering this mathematical challenge!

Key Takeaways: Mastering Hyperbola Equations

Before we wrap up, let's recap the key steps we took to solve this problem. These steps are a roadmap for tackling similar hyperbola problems in the future. Remember these, and you'll be a hyperbola equation whiz in no time!

1. Determine the Orientation: Is the hyperbola opening horizontally or vertically? This dictates which standard form equation you'll use. Look at the positions of the center, focus, and vertices to figure this out.
2. Identify the Center (h, k): The center is your starting point. It's the (h, k) in the standard form equation. Sometimes it's given directly, and sometimes you need to find it using other information.
3. Find the Value of 'a': 'a' is the distance between the center and a vertex. It's a key parameter that helps define the shape of the hyperbola.
4. Find the Value of 'c': 'c' is the distance between the center and a focus. It's related to a and b by the equation $c^2 = a^2 + b^2$.
5. Find the Value of 'b': Use the relationship $c^2 = a^2 + b^2$ to solve for b. This completes the set of parameters needed for the equation.
6. Write the Equation in Standard Form: Plug the values of h, k, a, and b into the appropriate standard form equation (horizontal or vertical). Simplify, and you've got your answer!

Practice Makes Perfect: Sharpen Your Skills

Now that we've walked through this problem together, the best way to solidify your understanding is to practice! Try working through similar problems with different centers, foci, and vertices. You can even create your own hyperbola challenges and solve them. The more you practice, the more comfortable you'll become with these concepts. Remember, math is like a muscle – the more you exercise it, the stronger it gets!

Conclusion: You're a Hyperbola Hero!

Well done, mathletes! We've successfully navigated the world of hyperbolas and found the equation of a hyperbola given its center, focus, and vertex. You've learned how to identify key parameters, use the standard form equations, and apply the relationship between a, b, and c. You're now well-equipped to tackle any hyperbola equation that comes your way. Keep exploring, keep learning, and keep enjoying the beauty of mathematics!