0For JEE Mains, ellipse notes should cover the standard equation, properties (focus, directrix, eccentricity, latus rectum), and parametric representation. Key formulas include the equation x²/a² + y²/b² = 1 (assuming a > b), and the relationships between a, b, and eccentricity e (b² = a²(1-e²)). Also, understanding the concepts of major and minor axes, vertices, and the relationship between the ellipse and its auxiliary circle are crucial.
Key Concepts and Formulas:
- Standard Equation: x²/a² + y²/b² = 1 (assuming a > b).
- Eccentricity (e): e = √(a² - b²) / a or e = c/a, where c is the distance from the center to each focus.
- Foci: (±ae, 0).
- Directrices: x = ±a/e.
- Latus Rectum: Length = 2b²/a.
- Major and Minor Axes: Lengths are 2a and 2b, respectively.
- Vertices: (±a, 0).
- Parametric Representation: (a cosθ, b sinθ).
- Sum of Focal Distances: For any point P on the ellipse, SP + S'P = 2a.
Important Properties:
- Eccentricity: 0 < e < 1.
- Auxiliary Circle: The circle described on the major axis as diameter, with equation x² + y² = a².
