The given equation is the standard form of an ellipse:

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

### Analyzing the Equation:

This is an equation of an ellipse centered at the origin \( (0, 0) \), with the following properties:

1. **Center**: \( (0, 0) \)
2. **Semi-major axis**: 
   - The larger denominator, 36, is under \( y^2 \), so the semi-major axis is along the \( y \)-axis.
   - The length of the semi-major axis is \( \sqrt{36} = 6 \).
3. **Semi-minor axis**:
   - The smaller denominator, 9, is under \( x^2 \), so the semi-minor axis is along the \( x \)-axis.
   - The length of the semi-minor axis is \( \sqrt{9} = 3 \).
4. **Vertices**:
   - Along the \( y \)-axis: \( (0, 6) \) and \( (0, -6) \)
   - Along the \( x \)-axis: \( (3, 0) \) and \( (-3, 0) \)

### General form of an ellipse equation:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Where:
- \( a \) is the length of the semi-minor axis (3),
- \( b \) is the length of the semi-major axis (6).

Would you like a more detailed explanation or visualization? 

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Here are some related questions for further exploration:

1. What is the distance between the foci of the ellipse?
2. How do you calculate the eccentricity of the ellipse?
3. Can you write the equation of a horizontal ellipse?
4. How does this equation change if the ellipse is shifted from the origin?
5. How would the equation look if the ellipse were a circle?

**Tip**: The semi-major axis is always associated with the larger denominator in the ellipse equation.

Explore the geometry of the ellipse equation x^2/9 + y^2/36 = 1. Learn about the center, axes, vertices, and how to interpret the equation in standard form. This solution includes step-by-step analysis of the semi-major and semi-minor axes. Ideal for high school students in Grades 10-12.

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