Conic Section Ellipse Class 11 Notes & Numericals for IIT JEE

The standard equation of an ellipse with centre at the origin is x²/a² + y²/b² = 1, where a > b > 0. If the major axis is along the x-axis, a² appears under x². If the major axis is along the y-axis, a² appears under y². This form is derived directly from the definition that the sum of focal distances equals 2a.

The latus rectum is a chord drawn through a focus of the ellipse, perpendicular to the major axis. Its length is 2b²/a, where a is the semi-major axis and b is the semi-minor axis. Each ellipse has two latus recta (one per focus). This value is a high-frequency formula in JEE Main — memorise it with a derivation for best retention.

Eccentricity of an ellipse is e = c/a, where c = √(a² − b²). For an ellipse, eccentricity always satisfies 0 < e < 1. When e = 0, the ellipse is a perfect circle. As e approaches 1, the ellipse becomes increasingly elongated. This range (0, 1) is the defining characteristic that separates an ellipse from other conics.

In an ellipse, c² = a² − b², where a is the semi-major axis, b is the semi-minor axis, and c is the distance from the centre to each focus. This can also be written as b² = a²(1 − e²). This relationship is the foundation of almost every ellipse derivation and problem in Class 11 and JEE.

Based on NTA's official paper patterns from 2021–2024, at least one question from the ellipse appears in almost every JEE Main session. The full Conic Sections chapter typically contributes 1–2 questions per paper (4–8 marks). Ellipses, parabolas, and hyperboloids are equally important within this chapter.

A focal chord is any chord of an ellipse that passes through one of its foci. The latus rectum is the shortest focal chord and is perpendicular to the major axis. If a focal chord makes an angle with the major axis, its length can be calculated using the focal chord length formula: L = 2b²/a · 1/(1 − e²cos²θ), where θ is the angle with the major axis.

A circle is a special case of an ellipse where a = b, making eccentricity e = 0. In a circle, both foci coincide at the centre and the radius is uniform in all directions. In an ellipse, a ≠ b, the two foci are distinct, and the shape is stretched along the major axis. The equation of a circle x² + y² = r² is x²/r² + y²/r² = 1, fitting the ellipse form with a = b = r