What is the generalised equation for a conic

The generalized equation for a conic section is:

\[
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
\]

where:
- \( A \), \( B \), and \( C \) determine the type of conic section.
- \( D \) and \( E \) control the translation (shift) of the conic.
- \( F \) is the constant term affecting the conic's size or location.

Different types of conic sections arise based on the values of \( A \), \( B \), and \( C \):
1. **Circle**: \( A = C \) and \( B = 0 \)
2. **Ellipse**: \( A \neq C \), \( A > 0 \), \( C > 0 \), and \( B = 0 \)
3. **Parabola**: Either \( A = 0 \) or \( C = 0 \) (but not both)
4. **Hyperbola**: \( A \) and \( C \) have opposite signs (i.e., \( A \cdot C < 0 \))
5. **Degenerate Conic**: Special cases where the equation represents a point, a line, or two intersecting lines.

The discriminant \( \Delta = B^2 - 4AC \) helps classify the conic:
- \( \Delta < 0 \): Ellipse or circle.
- \( \Delta = 0 \): Parabola.
- \( \Delta > 0 \): Hyperbola.

Would you like details on how each conic section is derived from this equation?

### 5 Related Questions:
1. How is the equation of a circle derived from the general form?
2. What role does the discriminant \( \Delta = B^2 - 4AC \) play in identifying conics?
3. How do translations affect the conic section equation?
4. What are degenerate conics, and how are they represented?
5. How can rotation of axes simplify the general conic equation?

### Tip:
Understanding how to manipulate and classify the generalized conic equation can help solve geometric problems involving curves and surfaces.

What is the generalised equation for a conic?

Understand the generalized equation for conic sections and learn how to classify them based on the discriminant. This guide explores conics such as ellipses, parabolas, and hyperbolas, providing insight into their algebraic structure. Suitable for Grades 10-12 students.

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