Parabola for JEE Main 2026: Complete Exam Guide

The parabola is the simplest conic section and typically the starting point for JEE Main coordinate geometry. JEE contributes two to three marks from parabola questions every session, testing the focus-directrix definition, tangent and normal equations, chords, and the parametric representation. The good news: parabola formulae are the cleanest among the three non-degenerate conics, and once you have mastered the standard results, the question types become highly predictable.

Standard Forms and Key Parameters

The standard parabola y² = 4ax (a > 0) opens to the right, with focus at (a, 0), directrix x = −a, vertex at the origin, and latus rectum length 4a. The four orientations and their parameters: y² = 4ax (right, focus (a,0)); y² = −4ax (left, focus (−a,0)); x² = 4ay (up, focus (0,a)); x² = −4ay (down, focus (0,−a)). For a parabola in the form (y−k)² = 4a(x−h), the vertex is (h, k) and focus is (h+a, k). JEE uses these shifted forms regularly — identify the vertex first, then apply the standard formulae with the shifted coordinates.

Any point on y² = 4ax can be written parametrically as (at², 2at). This parametric form is the most powerful tool in parabola problems because equations of tangents and normals at parametric points take simple forms. Test your parabola formula speed with a free coordinate geometry mock.

Tangent and Normal: Standard Results

Tangent to y² = 4ax at (x₁, y₁): yy₁ = 2a(x + x₁). At parametric point (at², 2at): ty = x + at². Slope form: y = mx + a/m (tangent with slope m, valid for all m ≠ 0). The condition for y = mx + c to be tangent: c = a/m. This condition eliminates many "find the tangent from external point" problems — substitute the external point into y = mx + a/m and solve for m.

Normal to y² = 4ax at (at², 2at): y + tx = 2at + at³. Slope form: y = mx − 2am − am³. Normal is a cubic in the slope parameter — three normals can be drawn from an external point in general. The three normal slopes m₁, m₂, m₃ satisfy m₁ + m₂ + m₃ = 0, m₁m₂ + m₂m₃ + m₃m₁ = −2 − h/a, and m₁m₂m₃ = −k/a (by Vieta's formulas applied to the normal cubic). These relations let you find the sum or product of the slopes of the three normals without solving the cubic explicitly — a high-efficiency technique for JEE.

Chord of Contact and Polar

From an external point (h, k), the chord of contact (the chord joining the two points of tangency) to y² = 4ax has equation ky = 2a(x + h) — this is the T = 0 form with (x₁, y₁) replaced by (h, k). The polar of a point (h, k) with respect to the parabola is ky = 2a(x+h) — identical to the chord of contact for external points, but valid for any point.

Pairs of tangents from (h, k): the combined equation of the two tangent lines uses the relation T² = SS₁, where T = ky − 2a(x+h), S = y² − 4ax, S₁ = k² − 4ah. This SS₁ = T² formula is the unified result for the pair of tangents from any external point to a conic — it works the same for circle, parabola, ellipse, and hyperbola with only the definition of T and S changing. For the other conics, see our ellipse guide and hyperbola guide.