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To determine the type of conic section represented by the given equation \( 4x^2 - y^2 - 32x + 6y + 51 = 0 \), we need to analyze its general form and discriminant.

The general form of a conic section equation is:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]

Here, comparing with the given equation \( 4x^2 - y^2 - 32x + 6y + 51 = 0 \), we have:
- \( A = 4 \)
- \( B = 0 \)
- \( C = -1 \)
- \( D = -32 \)
- \( E = 6 \)
- \( F = 51 \)

The discriminant \( \Delta \) for a conic section is given by:
\[ \Delta = B^2 - 4AC \]

Substituting the values:
\[ \Delta = 0^2 - 4 \cdot 4 \cdot (-1) = 0 - 4 \cdot 4 \cdot (-1) = 16 \]

Since \( \Delta > 0 \), the conic section is a **hyperbola**.

Would you like more details or have any questions?

Here are some questions you can ask:
1. How do you classify conic sections using the discriminant?
2. Can you explain how to convert a conic section to its standard form?
3. How do you find the center of a hyperbola?
4. What are the asymptotes of a hyperbola?
5. Can you derive the equation of a hyperbola given its vertices and foci?

**Tip:** Always remember to compare your conic section equation to the general form and use the discriminant to classify it quickly.

Learn how to determine the type of conic section represented by the equation 4x^2 - y^2 - 32x + 6y + 51 = 0. This problem covers key concepts including the general form of conic sections, discriminant analysis, and classification of conic sections using discriminant. Suitable for students in Grades 11-12.

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