Parabola for JEE Main: Complete Guide

The Parabola is one of the three conic sections tested heavily in JEE Main mathematics, typically contributing 1–2 questions per session from among the four conics (parabola, ellipse, hyperbola, circle). Parabola problems in JEE Main test knowledge of the standard forms, parametric representation, tangent and normal equations, chord properties, and family of lines/curves. This guide provides systematic coverage of all subtopics with the exact formula derivations and shortcut results that allow efficient problem-solving under timed conditions.

Standard Equations and Properties

The four standard parabolas: (1) y² = 4ax (opens right, vertex at origin, focus at (a,0), directrix x = −a). (2) y² = −4ax (opens left). (3) x² = 4ay (opens up, focus at (0,a), directrix y = −a). (4) x² = −4ay (opens down). The vertex form for a shifted parabola: (y−k)² = 4a(x−h) with vertex at (h,k). Any parabola y = Ax² + Bx + C can be rewritten in vertex form by completing the square. Focus-directrix definition: the set of all points equidistant from focus and directrix. Length of latus rectum (chord through focus perpendicular to axis): 4a for all four standard forms. Focal length (distance from vertex to focus): a. Focal chord: any chord passing through the focus. Length of focal chord: for endpoints with parameters t1 and t2 (where x = at², y = 2at for y² = 4ax): length = a(t1 + t2)² + ... or directly, a(t1 − t2)² + (2at1 − 2at2)² = .... The specific formula: if a focal chord has parameter t, the other endpoint has parameter −1/t, and length = a(t + 1/t)² ≥ 4a (minimum is latus rectum). For the broader conic sections context, see our Conic Sections Complete Guide.

Parametric form for y² = 4ax: any point on the parabola can be written as (at², 2at) for parameter t ∈ R. The parameter t is the slope of the line joining the origin to the point? No — t is just a convenient parameter. The line joining two points (at1², 2at1) and (at2², 2at2) has slope 2/(t1+t2). The chord joining these two points has equation: y(t1+t2) = 2x + 2a·t1·t2. This is the parametric chord equation — memorise it directly, as it generates the tangent, normal, and chord of contact as special cases.

Tangent and Normal Equations

Tangent to y² = 4ax at point (at², 2at): equation is ty = x + at². Equivalently in slope form: if slope = m, tangent is y = mx + a/m (valid for m ≠ 0). Point of contact with slope m: (a/m², 2a/m). Normal at (at², 2at): equation is y = −tx + 2at + at³. In slope form: if slope of normal is m, normal is y = mx − 2am − am³; point of tangency on the curve: (am², −2am). Three normals can be drawn from an external point to y² = 4ax — the three parameters m1, m2, m3 satisfy: m1 + m2 + m3 = 0, m1·m2 + m2·m3 + m3·m1 = (2a−h)/a (where (h,k) is the external point), m1·m2·m3 = −k/a. These relations (Vieta's formulas for the cubic in m) generate JEE Main questions: "if two normals from point P are along the axes, find P." Practise parabola tangent and normal problems on our JEE Main math mock tests — these are among the most efficiently-solved problems in coordinate geometry once you know the slope form results.

Chord of contact: for tangents from external point (h,k) to y² = 4ax, the chord of contact has equation y·k = 2a(x+h). This is T = 0 in the standard T notation, where T = y·y1 − 2a(x+x1). Pair of tangents from external point (h,k): SS1 = T², where S = y² − 4ax, S1 = k² − 4ah, T = y·k − 2a(x+h). These results — T=0 and SS1 = T² — apply to all conics with appropriate substitution, making them one of the most powerful shortcut formulas in coordinate geometry.

Focal Chord Properties and Special Results

For a focal chord with endpoints P(at1², 2at1) and Q(at2², 2at2): t1·t2 = −1. This is the most important focal chord property — memorise it. Implications: if one endpoint has parameter t, the other has parameter −1/t. Length of focal chord = a(t − (−1/t))² = a(t + 1/t)² ≥ 4a. Minimum length occurs when t = ±1, giving the latus rectum (length 4a). Semi-latus rectum harmonic mean: 1/|PS| + 1/|QS| = 2/a (harmonic mean of the two focal distances equals 2/a), where |PS| and |QS| are distances from P and Q to the focus. This is a beautiful result tested in JEE Advanced and occasionally JEE Main. The geometric result: the tangents at the endpoints of a focal chord are perpendicular (since t1·t2 = −1 and slopes of tangents are 1/t1 and 1/t2, their product is −1). Therefore, tangents at ends of a focal chord meet on the directrix at 90°.

Reflection property of parabola: any ray parallel to the axis reflects through the focus. A ray from the focus reflects parallel to the axis. This property is used in parabolic mirrors (satellite dishes, telescopes, headlights) and is the conceptual foundation of many JEE Main questions about optics and parabolas combined. Pole and polar: polar of point (h,k) with respect to y² = 4ax is y·k = 2a(x+h) — same as chord of contact formula, but valid for any point (not just external). If P lies on the polar of Q, then Q lies on the polar of P (reciprocal property of pole and polar).

Director Circle and Family of Parabolas

For y² = 4ax, the director circle (locus of point from which perpendicular tangents can be drawn) is the directrix x + a = 0 — a degenerate circle. This is unique to parabolas; ellipses and hyperbolas have genuine circles as director circles. Chord with a given midpoint (h,k): equation is y·k − 2a(x+h) = k² − 4ah, i.e., T = S1. This is the "T = S1" formula and directly gives the chord equation without any other computation. JEE Main: "find the equation of chord of y² = 8x with midpoint (2,4)." Here a = 2: T = S1 gives y(4) − 4(x+2) = 16 − 16 = 0, so 4y − 4x − 8 = 0, y − x = 2. Create a free account on our platform for 150+ parabola practice problems with increasing difficulty levels. Our premium subscription includes full conic sections problem banks. For the ellipse and hyperbola that complete the conic sections chapter, see our Hyperbola and Ellipse Guide.

JEE Main parabola problems are almost always standard-type problems once you know the results. Build a parabola formula card with: all four standard forms, parametric equations, tangent in point and slope form, normal in point and slope form, chord of contact, midpoint chord, focal chord property (t1t2 = −1), and the T=0, SS1=T², T=S1 formulas. A 2-hour investment in this card and its associated 20 practice problems makes parabola a reliable 4-mark source on exam day.