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?

---

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.