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The equation given is:

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

To determine the type of conic section, let's rewrite the equation in a more standard form.

First, divide both sides of the equation by 36 to normalize the equation:

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

Simplify each term:

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

This is the standard form of the equation for an ellipse:

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

where \(a^2 = 9\) and \(b^2 = 4\).

Thus, the given equation represents an **ellipse**.

Would you like more details or have any questions about this solution?

Here are 5 related questions you might find helpful:
1. How do you determine the foci of an ellipse?
2. What is the standard form of a hyperbola?
3. How can you identify a parabola from its equation?
4. What are the major and minor axes of an ellipse?
5. How do you find the eccentricity of an ellipse?

**Tip:** When identifying conic sections, always try to rewrite the equation in its standard form. This will help you recognize the type of conic section more easily.

Understand how to identify the conic section of an equation through detailed analysis. This example covers the determination of an ellipse from the equation \( 4x^2 + 9y^2 = 36 \), including step-by-step normalization and interpretation of the standard form. Ideal for high school students and anyone interested in conic sections in mathematics.

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