Ellipse for JEE Main 2026: Complete Exam Guide

The ellipse is the most elegant of the conic sections and the one JEE Main tests with the greatest variety of question types: locus problems, tangent-normal conditions, reflection properties, and auxiliary circle relations. Mastering the ellipse also reinforces the hyperbola (by analogy, with sign changes) and provides geometric intuition that transfers to the other conics. This guide covers every formula and result the exam tests, with the sign conventions that separate correct answers from careless errors.

Standard Equation and Key Parameters

The standard ellipse x²/a² + y²/b² = 1 with a > b > 0 has foci at (±c, 0) where c² = a² − b² (major axis along x). Eccentricity e = c/a < 1. Semi-latus rectum = b²/a; full latus rectum = 2b²/a. Directrices at x = ±a/e = ±a²/c. The sum of focal radii SP + S'P = 2a for any point P on the ellipse — the focal-distance definition. This sum = 2a result is the fastest way to find the distance from a focus to a point on the ellipse: SP = a − ex and S'P = a + ex (where x is the x-coordinate of P).

The auxiliary circle has equation x² + y² = a². For any point P on the ellipse, draw a vertical line to meet the auxiliary circle at Q — the angle QCA (where C is the centre and A is the end of the major axis) is the eccentric angle θ: P = (a cosθ, b sinθ). Every ellipse calculation becomes clean in this parametric form. Take a free ellipse mock to test your formula recall speed.

Tangent, Normal, and Chord of Contact

Tangent at (x₁, y₁): xx₁/a² + yy₁/b² = 1. At (a cosθ, b sinθ): x cosθ/a + y sinθ/b = 1. Slope form: y = mx ± √(a²m² + b²). Unlike the hyperbola, this always exists for any m. The condition for y = mx + c to be tangent: c² = a²m² + b². This is the circle-tangent condition, generalised to the ellipse.

Normal at (x₁, y₁): a²x/x₁ − b²y/y₁ = a² − b². The normal at any point on the ellipse passes through neither focus in general, but the normal is harmonic-conjugate to the tangent with respect to the foci — a geometric fact behind the reflection property (elliptical mirrors focus light from one focus to the other). Chord of contact from external point (h, k): xh/a² + yk/b² = 1. Pair of tangents: T² = SS₁ as usual, where T = xh/a² + yk/b² − 1 and S₁ = h²/a² + k²/b² − 1.

Special Chords: Focal Chord and Chord with Midpoint

A focal chord passes through a focus. For focal chord through focus (c, 0) in the parametric form (a cosθ, b sinθ), the two ends satisfy specific parametric relations. The semi-latus rectum b²/a is the distance from the focus to the ellipse along the perpendicular to the major axis — this makes it the focal chord of minimum length. The harmonic mean of the two focal-chord segment lengths equals the semi-latus rectum — a fact that appears as a direct question.

Chord with midpoint (h, k): using T = S₁, the equation is xh/a² + yk/b² = h²/a² + k²/b². This "midpoint chord" result connects to the concept of diameters of conics (the locus of midpoints of parallel chords is a straight line, and conjugate diameters are related by the slope product rule: m₁m₂ = −b²/a² for conjugate diameters of the ellipse). For complete conic coverage, see our parabola guide, hyperbola guide, and circle and radical axis guide.