Ellipse Equation And Graphing Explained Center At (0,0)

Aug 2, 2025 by Rajiv Sharma 56 views

Finding the Equation and Graphing an Ellipse Centered at the Origin

Hey guys! Let's dive into the fascinating world of ellipses! Today, we're going to tackle a problem where we need to find the equation of an ellipse and then graph it. We're given some key information: the center of the ellipse is at (0,0), one focus is at (12,0), and one vertex is at (-15,0). Buckle up, because we're about to unravel this geometric puzzle!

Understanding the Ellipse

Before we jump into the calculations, let's make sure we're all on the same page about what an ellipse actually is. Think of it as a stretched-out circle. Unlike a circle, which has a single center, an ellipse has two foci (plural of focus). The sum of the distances from any point on the ellipse to the two foci is constant. This constant sum is a crucial piece of information that helps us define the ellipse mathematically.

Now, the major axis is the longest diameter of the ellipse, passing through both foci and the center. The vertices are the endpoints of the major axis. The minor axis, on the other hand, is the shortest diameter, perpendicular to the major axis and also passing through the center. The endpoints of the minor axis are called co-vertices.

Key parameters that define an ellipse are a, b, and c. 'a' represents the distance from the center to a vertex along the major axis, 'b' represents the distance from the center to a co-vertex along the minor axis, and 'c' represents the distance from the center to a focus. These parameters are related by the equation c^2 = a^2 - b^2, which we will use later to find the missing piece of our ellipse equation.

When the center of the ellipse is at the origin (0,0) and the major axis lies along the x-axis, the standard form of the ellipse equation is x^2/a2 + y^2/b2 = 1. If the major axis lies along the y-axis, the equation becomes x^2/b2 + y^2/a2 = 1. It's super important to remember that a is always greater than b.

Step 1: Identifying the Orientation and 'a'

Our first clue is the location of the focus at (12,0) and the vertex at (-15,0). Since both of these points lie on the x-axis, and the center is at (0,0), we know that the major axis of our ellipse lies along the x-axis. This means our ellipse will be wider than it is tall.

We're also given that a vertex is at (-15,0). The distance from the center (0,0) to this vertex is 15 units. Remember, 'a' is the distance from the center to a vertex, so we know that a = 15. This is a huge step forward!

Step 2: Finding 'c'

Next, we need to find the value of 'c', which is the distance from the center to a focus. We're given that one focus is at (12,0). The distance from the center (0,0) to this focus is 12 units. Therefore, c = 12.

Now that we have both 'a' and 'c', we're well on our way to finding the equation of the ellipse.

Step 3: Calculating 'b'

Here's where the relationship c^2 = a^2 - b^2 comes into play. We know 'a' is 15 and 'c' is 12, so we can plug these values into the equation and solve for 'b':

12^2 = 15^2 - b^2 144 = 225 - b^2 b^2 = 225 - 144 b^2 = 81 b = 9

So, we've found that b = 9. Now we have all the pieces we need to write the equation of the ellipse!

Step 4: Writing the Equation

We know the major axis lies along the x-axis, so the equation of the ellipse will be in the form x^2/a2 + y^2/b2 = 1. We've found that a = 15 and b = 9. Let's plug these values into the equation:

x^2/152 + y^2/92 = 1 x^2/225 + y^2/81 = 1

Therefore, the equation of the ellipse is x^2/225 + y^2/81 = 1.

Graphing the Ellipse

Now that we have the equation, let's bring this ellipse to life by graphing it!

Step 1: Plot the Center

The center of our ellipse is at (0,0), so that's our starting point.

Step 2: Plot the Vertices

Since a = 15 and the major axis is along the x-axis, the vertices are located at (±a, 0), which means they are at (15,0) and (-15,0). Plot these points.

Step 3: Plot the Co-vertices

Since b = 9, the co-vertices are located at (0, ±b), which means they are at (0,9) and (0,-9). Plot these points as well.

Step 4: Plot the Foci

We know c = 12, so the foci are located at (±c, 0), which means they are at (12,0) and (-12,0). Plot these points. While the foci aren't directly used to draw the ellipse, they are important for understanding its shape and properties.

Step 5: Sketch the Ellipse

Now, carefully sketch the ellipse by drawing a smooth, oval-shaped curve that passes through the vertices and co-vertices. The ellipse should be centered at (0,0) and stretched along the x-axis.

Remember, the ellipse should be symmetrical about both the x-axis and the y-axis. The distance from the center to each vertex is 15 units, and the distance from the center to each co-vertex is 9 units.

Key Features to Observe in Your Graph

The Major Axis: The horizontal line segment connecting the vertices (-15,0) and (15,0). Its length is 2*a = 30 units.
The Minor Axis: The vertical line segment connecting the co-vertices (0,-9) and (0,9). Its length is 2*b = 18 units.
The Foci: Located inside the ellipse, closer to the center than the vertices. They influence the “roundness” of the ellipse. The closer the foci are to the center, the more circular the ellipse appears.

Wrapping Up

We've successfully found the equation of the ellipse, which is x^2/225 + y^2/81 = 1, and we've learned how to graph it. By understanding the relationships between the center, vertices, co-vertices, foci, and the parameters a, b, and c, you can conquer any ellipse problem that comes your way. Keep practicing, and you'll become an ellipse expert in no time! Remember, the key is to break down the problem into smaller, manageable steps and to use the given information strategically. You got this, guys!