Conic Sections JEE Main: Parabola, Ellipse & Hyperbola

Conic sections is among the highest-weightage coordinate geometry topics in JEE Main Mathematics, contributing two to three questions per session. The chapter covers the parabola, ellipse, and hyperbola — three curves defined by the intersection of a plane with a double cone. JEE Main tests standard equations, parametric forms, tangents, normals, and chord-related results for all three conics. The techniques for all three share significant structural similarity, making it efficient to study them together and identify the parallel results.

The Parabola

The standard parabola y² = 4ax has focus at (a, 0), directrix x = -a, vertex at the origin, and axis along the x-axis. The parametric form is (at², 2at). JEE Main tests the equation of the tangent at a given point (or in parametric form), the equation of the normal, and properties of chords — particularly the focal chord (a chord through the focus) and the chord of contact.

The tangent at the point (at², 2at) on y² = 4ax is ty = x + at². The condition for a line y = mx + c to be tangent to y² = 4ax is c = a/m. The locus of the foot of perpendicular from the focus to a tangent to the parabola is the tangent at the vertex. These results and their analogues appear in JEE Main problems in both direct form and as intermediate steps in more complex geometric problems. Connect with our circles guide for the coordinate geometry tools that underlie conic section problems.

The Ellipse

The standard ellipse x²/a² + y²/b² = 1 (with a > b) has semi-major axis a, semi-minor axis b, and eccentricity e = √(1 - b²/a²). The two foci are at (±ae, 0). The directrices are at x = ±a/e. The sum of distances from any point on the ellipse to the two foci is constant and equal to 2a — the fundamental definition of the ellipse in terms of foci.

The parametric form is (a cos θ, b sin θ). The equation of the tangent at the parametric point is (x cos θ)/a + (y sin θ)/b = 1. JEE Main tests the relationship between the tangent and normal to the ellipse, the chord of contact from an external point, and the locus of the midpoint of chords with a given slope. The auxiliary circle of the ellipse (the circle with the major axis as diameter) is useful for relating the eccentric angle to the geometry of the ellipse. Take a free mock test on conic sections to practice tangent and normal calculations.

The Hyperbola

The standard hyperbola x²/a² - y²/b² = 1 has eccentricity e = √(1 + b²/a²) > 1. The two foci are at (±ae, 0), and the two directrices are at x = ±a/e. The asymptotes are y = ±(b/a)x — lines that the hyperbola approaches but never crosses. The absolute difference of distances from any point on the hyperbola to the two foci is constant and equal to 2a.

The rectangular hyperbola xy = c² is a special case with perpendicular asymptotes and is heavily tested in JEE Main because it has the simplest parametric form (ct, c/t). The tangent at (ct, c/t) is x/t + yt = 2c, and the normal at this point passes through the center. The chord joining points (ct₁, c/t₁) and (ct₂, c/t₂) on the rectangular hyperbola has equation x + t₁t₂y = c(t₁ + t₂), and many elegant chord properties follow from the simplicity of this parametric form.

Common Conic Section Techniques

Across all three conics, the T = 0 substitution gives the tangent at a known point, the chord of contact from an external point, and the equation of the chord whose midpoint is known. Mastering this unified substitution technique — and understanding when to apply each form — is one of the most efficient approaches to JEE Main conic problems, because you can solve three types of problems with one unified formula structure rather than three separate ones.

For a complete coordinate geometry strategy, see our coordinate geometry guide. This chapter connects to 3D geometry through the analogous surfaces: our 3D geometry guide builds naturally on the coordinate geometry foundations here. Follow the 30-day math plan and sign up free for our 250+ conic sections problem bank.