91Ó°ÊÓ

An ellipse is a curved shape like a squished circle, defined by its equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). In this equation, \(a^2\) and \(b^2\) are constants representing the distances squared from the center to the vertex along the x-axis and y-axis, respectively. Here, \(a\) and \(b\) are positive numbers representing the lengths of the semi-major and semi-minor axes,
with \(a\) being the distance along the x-axis and \(b\) along the y-axis.

For example, given the equation \(\frac{x^2}{4} + \frac{y^2}{25} = 1\), we identify that \(a^2 = 4\) and \(b^2 = 25\), and can calculate that \(a = 2\) and \(b = 5\). This tells us that the ellipse stretches 2 units along the x-axis and 5 units along the y-axis from its center.

Vertices and co-vertices are special points on the ellipse that help in defining its shape and dimensions.

In our example equation \(\frac{x^2}{4} + \frac{y^2}{25} = 1\), we know \(a = 2\) and \(b = 5\).
The vertices are the points along the major axis (x-axis for this case) and are located at \(\text{a}= 2\) units from the center (0,0).
This gives us the vertices at:

• (2, 0)
• (-2, 0)

The co-vertices are the points along the minor axis (y-axis for this case) and are located \(\text{b}= 5\) units from the center.
This gives us the co-vertices at:

• (0, 5)
• (0, -5)

Plotting an ellipse on a coordinate plane becomes straightforward once we know the center, vertices, and co-vertices.

Step-by-step plotting guide:

• Mark the center point on the coordinate plane. In our example, it's at (0, 0).
• Then, plot the vertices and co-vertices points identified. For our example, vertices are at (2, 0) and (-2, 0),
  and co-vertices are at (0, 5) and (0, -5).
• Once the key points are plotted, draw a smooth curve passing through these points to form the ellipse.

By following these steps and using the identified points as guides,
you can create an accurate representation of the ellipse on a graph.