MathsMediumClass 12

Standard Equation — x²/a² + y²/b² = 1

Conic Sections

JEE Qs

Hard

min

Master the relationships between 'a', 'b', 'e', and the focal property, as these are frequently tested in conceptual and problem-solving questions.

🧮 Key Formulas

x^2/a^2 + y^2/b^2 = 1 (for ellipse with center at origin, a > b)

Eccentricity: e = sqrt(1 - b^2/a^2)

Relation between a, b, e: b^2 = a^2(1 - e^2)

Foci coordinates: (±ae, 0)

Vertices coordinates: (±a, 0)

Co-vertices coordinates: (0, ±b)

Equations of directrices: x = ±a/e

Length of major axis: 2a

Length of minor axis: 2b

Length of latus rectum: 2b^2/a

Focal property: PF1 + PF2 = 2a (for any point P on the ellipse and foci F1, F2)

✅ Key Points for JEE

1Identify 'a' as the semi-major axis length and 'b' as the semi-minor axis length. The major axis is along the x-axis if `a^2` is under `x^2` and `a > b`; it's along the y-axis if `a^2` is under `y^2` and `a > b` (then the foci and vertices switch axes).
2Eccentricity 'e' (0 < e < 1) indicates the ellipse's 'ovalness'; closer to 0 means more circular, closer to 1 means more elongated.
3All key features (foci, vertices, directrices, latus rectum) are directly derived from 'a', 'b', and 'e'. Understanding their interrelationships is more important than rote memorization.
4The focal property, `PF1 + PF2 = 2a`, is a fundamental definition and useful for solving problems involving distances from foci.

⚠️ Common Mistakes

✕Incorrectly identifying 'a' and 'b' from the equation: 'a' always represents the semi-major axis length. If `x^2/A + y^2/B = 1`, then `a = max(sqrt(A), sqrt(B))` and `b = min(sqrt(A), sqrt(B))`. This determines the orientation.
✕Using the eccentricity formula `e = sqrt(1 - (smaller denominator)/(larger denominator))` instead of `e = sqrt(1 - b^2/a^2)` where 'a' is explicitly the semi-major axis length.
✕Confusing the coordinates and equations for a horizontal major axis vs. a vertical major axis (e.g., placing foci on y-axis when major axis is horizontal).

📝 Practice Questions

See all

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

2019·MCQHard

Q69.If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P , then the distance of P from the origin (units), is: + (1) 2(3 2√2) (2) 3 + 2√2 + (3) √2 + 1 (4) 2(√2 1) JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper

2017·Multi conceptHard

NCERT Chapters

Class 11 Maths Ch 11: Conic Sections