Hyperbola for JEE Main 2026: Complete Guide

Hyperbola is the most algebraically complex conic section, but JEE Main tests it within a well-defined set of question types: eccentricity calculations, equation of asymptotes, tangent and normal conditions, and the rectangular hyperbola. Unlike the ellipse, where intuition from circles helps, the hyperbola requires careful attention to signs in every formula — the main source of student errors here. This guide presents every formula with sign emphasis and covers all exam-relevant results.

Standard Equation and Key Parameters

The standard hyperbola x²/a² − y²/b² = 1 has foci at (±c, 0) where c² = a² + b², eccentricity e = c/a > 1. The transverse axis (real axis) has length 2a; the conjugate axis has length 2b (unlike the ellipse, a > b is not required — a and b are independent). The relation c² = a² + b² (note the + sign, unlike the ellipse where c² = a² − b²) is the most important difference from ellipse formulae to memorise.

Directrices are at x = ±a/e = ±a²/c. Latus rectum length = 2b²/a. Parametric representation: x = a secθ, y = b tanθ (or x = a coshθ, y = b sinhθ for the hyperbolic parametrisation). JEE frequently asks for eccentricity given the ratio a:b or given the angle between asymptotes — having e = c/a and c² = a² + b² available immediately resolves these. Take a free conic sections mock to test your hyperbola formula speed.

Asymptotes and Conjugate Hyperbola

The asymptotes of x²/a² − y²/b² = 1 are y = ±(b/a)x, or equivalently x/a ± y/b = 0. The asymptotes are the lines the hyperbola approaches but never touches. The angle between the asymptotes is 2tan⁻¹(b/a). A rectangular hyperbola has perpendicular asymptotes, requiring b = a, giving e = √2. The equation of a rectangular hyperbola can be written as xy = c² when the asymptotes are taken as the coordinate axes — this form is tested in "rectangular hyperbola" questions where you are asked for the tangent equation or the normal.

The conjugate hyperbola of x²/a² − y²/b² = 1 is −x²/a² + y²/b² = 1 (or y²/b² − x²/a² = 1). The pair of hyperbola and conjugate hyperbola share the same asymptotes. A key relation: the combined equation of the asymptotes is x²/a² − y²/b² = 0, which is the average of the hyperbola and conjugate hyperbola equations (x²/a² − y²/b² = 1 and −1 average to 0).

Tangent and Normal to a Hyperbola

Tangent at point (x₁, y₁) on x²/a² − y²/b² = 1: xx₁/a² − yy₁/b² = 1. This is the T = 0 shorthand (replace x² → xx₁, y² → yy₁). At parametric point (a secθ, b tanθ): tangent is x secθ/a − y tanθ/b = 1. Slope form of tangent: y = mx ± √(a²m² − b²), which exists only when a²m² > b² — unlike the ellipse where a²m² + b² always exists. This existence condition is tested directly.

Normal at (x₁, y₁): a²x/x₁ + b²y/y₁ = a² + b². A crucial property — the combined equation of any chord of contact, pair of tangents, or chord with midpoint uses the same T = S₁ and T² = SS₁ format as for the ellipse and parabola, with only the signs in T reflecting the hyperbola equation. For the complete conic sections coverage, combine this guide with our parabola guide, ellipse guide, and circle guide.