https://file.aiquickdraw.com/mathbot/file/1727778429928nfymz3q3.jpg

The equation visible in the image is:

\[
4x^2 + 9y^2 - 36 = 0
\]

This is the equation of an ellipse in standard form. To rewrite it in a clearer format, we can divide the entire equation by 36 to simplify it:

\[
\frac{4x^2}{36} + \frac{9y^2}{36} = \frac{36}{36}
\]

\[
\frac{x^2}{9} + \frac{y^2}{4} = 1
\]

This is the standard form of an ellipse equation:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Where \( a^2 = 9 \) and \( b^2 = 4 \), meaning that \( a = 3 \) and \( b = 2 \).

Thus, the ellipse has:
- Semi-major axis \( a = 3 \)
- Semi-minor axis \( b = 2 \)
- The center of the ellipse is at \( (0, 0) \), since there are no terms involving \( x \) or \( y \) shifts.

Let me know if you would like further details on ellipses or how to graph this equation.

Here are five related questions to explore further:
1. How do you find the foci of an ellipse?
2. What happens if the signs of \( x^2 \) and \( y^2 \) are different in the ellipse equation?
3. How can the equation of a circle be derived from the equation of an ellipse?
4. How does changing the coefficients of \( x^2 \) and \( y^2 \) affect the shape of the ellipse?
5. What are the standard forms of hyperbola equations, and how do they differ from ellipses?

**Tip**: Always divide through by the constant term in the equation when simplifying conic sections like ellipses or hyperbolas to easily compare it to the standard form.

Learn how to simplify and solve the ellipse equation 4x^2 + 9y^2 - 36 = 0. This step-by-step guide explains how to rewrite the equation into its standard form and discusses important concepts like the semi-major axis, semi-minor axis, and the center of the ellipse. Suitable for Grades 10-12.

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