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To identify the type of conic section each equation represents, we need to analyze the general forms and coefficients of the \(x^2\) and \(y^2\) terms. Let's break it down for each equation:

1. **\(x^2 = -2y^2 + 8\)**  
   This equation can be rewritten as \(x^2 + 2y^2 = 8\). Since the coefficients of \(x^2\) and \(y^2\) have opposite signs, this is a **hyperbola**.

2. **\(2x^2 + 4y^2 - 3x = 14\)**  
   This equation has both \(x^2\) and \(y^2\) with the same signs, indicating that it is an **ellipse** (if both are positive). The linear term \( -3x \) indicates a translation of the conic.

3. **\(4x^2 + x - 18y^2 + 12y = 5\)**  
   Here, \(x^2\) and \(y^2\) have opposite signs, so it’s a **hyperbola**.

4. **\(-x^2 + 8x - y = -4y^2\)**  
   This can be rewritten as \(-x^2 + 4y^2 + 8x = y\). Again, opposite signs for \(x^2\) and \(y^2\) indicate a **hyperbola**.

5. **\(3x^2 - 15y^2 + 2x - 3y = 4\)**  
   Opposite signs for \(x^2\) and \(y^2\), so this is a **hyperbola**.

6. **\(5x^2 + 5y - 8 = -5y^2 - 2x\)**  
   This simplifies into an equation where \(x^2\) and \(y^2\) terms have opposite signs. Thus, it's a **hyperbola**.

7. **\(-x^2 + 5y^2 + 3y = 5\)**  
   This has opposite signs for \(x^2\) and \(y^2\), confirming that it is a **hyperbola**.

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Would you like further details on each conic, or need assistance with another part of the problem? Here are some related questions:

1. How do you classify conic sections based on their general equation?
2. What are the key characteristics of a hyperbola?
3. How does translating a conic section affect its equation?
4. What is the difference between an ellipse and a circle in terms of their equation?
5. Can a linear term in a conic section equation affect the type of conic?

**Tip:** Always look at the signs and coefficients of the squared terms (\(x^2\) and \(y^2\)) to classify conic sections.

Identify which conic section each equation represents.

Learn how to identify conic sections such as ellipses, hyperbolas, and parabolas from quadratic equations. Analyze the signs and coefficients of x^2 and y^2 terms. Suitable for high school students in Grades 10-12.

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