https://file.aiquickdraw.com/mathbot/file/1726767010145b4ebg9q8.jpg

Let me analyze the image to identify the conic sections each equation represents.

### List of equations:
1. \( x^2 = -2y^2 + 8 \)
2. \( 2x^2 + 3x + 2y^2 + 2y = 42 \)
3. \( -x^2 + 5y^2 + 3y = 5 \)
4. \( 4x^2 + 3y^2 = 4x - 2y + 12 \)
5. \( -x^2 + 8x - y = -4y^2 \)
6. \( 5x^2 + 5y - 8 = -5y^2 - 2 \)
7. \( 2x^2 + 4y^2 - 3x = 14 \)

Now, I will identify the conic sections.

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### Identification of Conic Sections:

1. **\( x^2 = -2y^2 + 8 \)**
   - This is a **hyperbola** because there are two squared terms with opposite signs (\(x^2\) and \(-2y^2\)).

2. **\( 2x^2 + 3x + 2y^2 + 2y = 42 \)**
   - Both squared terms have the same coefficients, suggesting it is an **ellipse**. The linear terms (\(3x\) and \(2y\)) are not relevant for determining the conic type.

3. **\( -x^2 + 5y^2 + 3y = 5 \)**
   - Since one squared term is negative, this is a **hyperbola**.

4. **\( 4x^2 + 3y^2 = 4x - 2y + 12 \)**
   - Both squared terms are positive, and the equation is quadratic in both \(x\) and \(y\), making this an **ellipse**.

5. **\( -x^2 + 8x - y = -4y^2 \)**
   - Again, the squared terms have opposite signs (\(-x^2\) and \(-4y^2\)), which makes this a **hyperbola**.

6. **\( 5x^2 + 5y - 8 = -5y^2 - 2 \)**
   - This has opposite signs for the squared terms, indicating a **hyperbola**.

7. **\( 2x^2 + 4y^2 - 3x = 14 \)**
   - Both squared terms are positive, indicating this is an **ellipse**.

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Would you like more details or have any questions? Here are some related questions:

1. How can I classify conic sections from general equations?
2. What are the properties of ellipses and how do they differ from hyperbolas?
3. How do I convert a general conic equation into standard form?
4. What is the difference between a parabola and a hyperbola in terms of their equation structure?
5. How do I find the center of an ellipse or hyperbola from an equation?

**Tip:** Remember that the sign and coefficients of the squared terms in a conic equation help you identify whether it's a circle, ellipse, hyperbola, or parabola!

Identify which conic section each equation represents.

Explore how to identify conic sections such as ellipses, hyperbolas, and parabolas from algebraic equations. This guide provides a clear breakdown of conic sections identification using standard forms and discriminants, suitable for high school students.

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