The problem in the image asks for the equation of an ellipse centered at the origin, with given vertices and co-vertices. Here's the solution step-by-step based on the provided details:

Step 1: Recognize the standard form

The standard form of an ellipse centered at the origin is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Where:

• $a$ is the semi-major axis (the distance from the center to a vertex along the major axis),
• $b$ is the semi-minor axis (the distance from the center to a co-vertex along the minor axis).

Step 2: Identify the values for $a$ and $b$

From the image:

• One vertex is at $(-6, 0)$ and the other at $(6, 0)$. This means the major axis is horizontal, and the distance between the center and each vertex is 6 units. Therefore, $a = 6$.
• One co-vertex is at $(0, 3)$, and the other at $(0, -3)$. This means the minor axis is vertical, and the distance from the center to each co-vertex is 3 units. Therefore, $b = 3$.

Step 3: Plug values into the standard form

We now know:

• $a = 6$,
• $b = 3$.

So the equation of the ellipse becomes:

$\frac{x^2}{6^2} + \frac{y^2}{3^2} = 1$

Simplifying this:

$\frac{x^2}{36} + \frac{y^2}{9} = 1$

Final Equation:

The equation of the ellipse is:

$\boxed{\frac{x^2}{36} + \frac{y^2}{9} = 1}$

Let me know if you'd like further details on this problem! Here are some related questions for you to explore:

1. How can you determine the orientation of an ellipse from its equation?
2. What is the relationship between the foci and the major axis of an ellipse?
3. How do you identify the equation of an ellipse when the center is not at the origin?
4. What is the difference between an ellipse and a hyperbola in terms of their equations?
5. How do the lengths of the axes affect the shape of the ellipse?

Tip: For ellipses, the larger axis (major axis) determines the direction in which the ellipse stretches. It is important to know if the major axis is horizontal or vertical when setting up the equation!