Ellipse Equation: Center (3, -1), Minor Axis 2, Vertex (3, -10)

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Hey guys! Let's dive into the fascinating world of ellipses and figure out how to find their equations. Today, we're tackling a specific problem: finding the equation of an ellipse given its center, the length of its minor axis, and the coordinates of a vertex. Don't worry, it might sound complicated, but we'll break it down into easy-to-follow steps. So, grab your pencils and let's get started!

Understanding the Basics of Ellipses

Before we jump into the calculations, let's make sure we're all on the same page about what an ellipse actually is. An ellipse is essentially a stretched circle. Think of it like a circle that's been squashed or pulled in one direction. This shape has some key features that we need to understand:

  • Center: The central point of the ellipse. It's like the heart of the ellipse.
  • Major Axis: The longest diameter of the ellipse. It passes through the center and the two vertices.
  • Vertices: The endpoints of the major axis. These are the points furthest away from the center along the ellipse.
  • Minor Axis: The shortest diameter of the ellipse. It's perpendicular to the major axis and also passes through the center.
  • Co-vertices: The endpoints of the minor axis.
  • Foci: Two special points inside the ellipse that are used in its formal definition. We won't delve too deeply into foci here, but it's good to know they exist.

The standard equation of an ellipse depends on whether the major axis is horizontal or vertical:

  • Horizontal Major Axis: (x-h)²/a² + (y-k)²/b² = 1
  • Vertical Major Axis: (x-h)²/b² + (y-k)²/a² = 1

Where:

  • (h, k) is the center of the ellipse.
  • a is the distance from the center to a vertex (the semi-major axis).
  • b is the distance from the center to a co-vertex (the semi-minor axis).

Understanding these basics is crucial for tackling our problem. Now that we've got the groundwork laid, let's apply this knowledge to our specific scenario.

Identifying the Key Information

Okay, let's revisit the problem. We need to find the equation of an ellipse given the following information:

  • Center: (3, -1)
  • Minor Axis Length: 2
  • Vertex: (3, -10)

The first step is to extract the important pieces of information from this data. The center is a straightforward (h, k) value, which is essential for our equation. The minor axis length gives us information about the 'b' value in our equation, but we'll need to do a little calculation to get there. The vertex, along with the center, helps us determine the orientation of the ellipse and the 'a' value. Let's break it down further:

  • Center (h, k): We know this is (3, -1), so h = 3 and k = -1. This is a direct plug-in for our standard equation.
  • Minor Axis Length: The minor axis has a length of 2. Remember that the minor axis length is 2b, where 'b' is the distance from the center to a co-vertex. Therefore, 2b = 2, which means b = 1. This is another crucial piece of the puzzle.
  • Vertex: We have a vertex at (3, -10). Since the center is at (3, -1) and the vertex is at (3, -10), they share the same x-coordinate. This tells us that the major axis is vertical. Why? Because the vertices lie along the major axis, and in this case, the change in the y-coordinate is what defines the major axis direction.

Now, let's find the value of 'a'. Remember, 'a' is the distance from the center to a vertex. We can calculate this distance using the distance formula, but in this case, it's simpler. Since the x-coordinates are the same, we just need to find the difference in the y-coordinates:

| -10 - (-1) | = | -10 + 1 | = | -9 | = 9

So, a = 9. We've now found all the crucial values: h, k, a, and b. We're ready to plug them into the standard equation!

Determining the Orientation and Standard Equation

As we deduced earlier, since the center and the vertex have the same x-coordinate, the major axis is vertical. This means we'll be using the standard equation for an ellipse with a vertical major axis:

(x-h)²/b² + (y-k)²/a² = 1

This equation is the framework for our solution. It tells us how the x and y coordinates relate to the center, the semi-major axis (a), and the semi-minor axis (b). The key here is that the larger denominator is under the (y-k)² term, indicating a vertical stretch. If the larger denominator were under the (x-h)² term, it would indicate a horizontal stretch.

Now, it's time to substitute the values we found in the previous step into this equation. This is where all our hard work pays off! We have:

  • h = 3
  • k = -1
  • a = 9
  • b = 1

Plugging these values in, we get:

(x-3)²/1² + (y-(-1))²/9² = 1

Simplifying this, we have:

(x-3)²/1 + (y+1)²/81 = 1

And that's it! We've found the equation of the ellipse.

Plugging in the Values and Simplifying

Let's recap. We've determined that the equation of our ellipse is:

(x-3)²/1 + (y+1)²/81 = 1

This equation perfectly describes the ellipse with a center at (3, -1), a minor axis of length 2, and a vertex at (3, -10). Let's take a moment to appreciate what we've done. We started with some given information, carefully analyzed the relationships between the ellipse's features, and systematically plugged those values into the correct standard equation. That's problem-solving at its finest!

We can further simplify the equation by recognizing that 1² is simply 1, so we can write:

(x-3)² + (y+1)²/81 = 1

This is the final, simplified form of the equation. It clearly shows the center (3, -1), the semi-minor axis of 1 (since b² = 1), and the semi-major axis of 9 (since a² = 81).

Graphing the Ellipse (Optional)

While finding the equation is the main goal, visualizing the ellipse can help solidify your understanding. If you want to take it a step further, you can graph the ellipse. To do this, you'd:

  1. Plot the center at (3, -1).
  2. Since the major axis is vertical and a = 9, move 9 units up and 9 units down from the center to find the vertices. One vertex is already given at (3, -10), and the other would be at (3, 8).
  3. Since the minor axis has a length of 2 and b = 1, move 1 unit left and 1 unit right from the center to find the co-vertices. These would be at (2, -1) and (4, -1).
  4. Sketch the ellipse using these four points as a guide. It should be a smooth, oval shape centered at (3, -1), stretched vertically.

Graphing is a great way to double-check your work and make sure your equation makes sense visually.

Conclusion: Mastering Ellipse Equations

So, there you have it! We've successfully found the equation of an ellipse given its center, minor axis length, and a vertex. The key takeaways from this exercise are:

  • Understand the basic properties of an ellipse: center, major axis, minor axis, vertices, and co-vertices.
  • Identify the orientation of the ellipse (horizontal or vertical) based on the given information.
  • Choose the correct standard equation based on the orientation.
  • Carefully extract the values of h, k, a, and b from the given information.
  • Substitute these values into the standard equation and simplify.

Finding the equation of an ellipse might seem daunting at first, but by breaking it down into smaller steps and understanding the underlying principles, you can conquer any ellipse equation problem. Keep practicing, and you'll become an ellipse equation master in no time! Remember, the more you practice, the more confident you'll become. And hey, if you get stuck, just revisit these steps, and you'll be back on track in no time. Happy solving, guys!