# Parametric Forms Practice > Parametric forms practice in JEE Mathematics: parametric points on parabola, ellipse and hyperbola, focal chord conditions and tangent-intersection loci. - Canonical URL: https://prepelephant.com/topics/jee/mathematics/parametric-forms-practice - Exam / course: JEE · Subject: Mathematics - Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team - First published: 2026-10-02 - Last updated: 2026-10-02 - How to cite: "Parametric Forms Practice", PrepElephant, https://prepelephant.com/topics/jee/mathematics/parametric-forms-practice ## Direct answer One letter replaces two coordinates on a conic: (at², 2at) is any point of y² = 4ax, (a cosθ, b sinθ) any point of the ellipse, (a secθ, b tanθ) any point of the hyperbola — and most locus questions collapse to one-variable algebra in that letter. The parabola carries the heaviest kit: tangent at t is ty = x + at², normal is y = −tx + 2at + at³, chord joining t₁ and t₂ is y(t₁ + t₂) = 2x + 2at₁t₂. Two conditions do most of the work: t₁t₂ = −1 makes the chord focal, and t₁t₂ = −4 makes the tangents at its ends meet on the directrix. ## What you must remember - **Parabola kit:** point (at², 2at); tangent ty = x + at²; normal y = −tx + 2at + at³; tangent slope 1/t, so normal slope t. - **Focal chord:** t₁t₂ = −1; its length is a(t + 1/t)² ≥ 4a, so the latus rectum is the shortest focal chord; tangents at its ends are perpendicular and meet on the directrix. - **Tangent intersection:** tangents at t₁ and t₂ meet at (at₁t₂, a(t₁ + t₂)) — the seed of nearly every locus problem on the parabola. - **Right angle at vertex:** the chord of contact subtending 90° at the vertex satisfies t₁t₂ = −4, placing the tangent intersection on x = −4a, the directrix. - **Ellipse and hyperbola:** ellipse point (a cosθ, b sinθ) with tangent (x/a)cosθ + (y/b)sinθ = 1; hyperbola point (a secθ, b tanθ) with tangent (x/a)secθ − (y/b)tanθ = 1. - **Rectangular hyperbola:** xy = c² is parametrised by (ct, c/t), tangent x/t + ty = 2c. - **Parameter meanings differ:** t on the parabola is a slope proxy, while θ on the ellipse is an eccentric angle — never transplant formulas between curves. ## A locus from two parameters Tangents to y² = 4ax at points t₁ and t₂ meet at P = (at₁t₂, a(t₁ + t₂)). Require the chord of contact — the chord through the two contact points — to subtend a right angle at the vertex O(0, 0). The slope of OQ₁, where Q₁ = (at₁², 2at₁), is 2at₁/at₁² = 2/t₁; likewise the slope of OQ₂ is 2/t₂. Perpendicularity of OQ₁ and OQ₂ gives (2/t₁)(2/t₂) = −1, i.e. t₁t₂ = −4. Substituting into P: the x-coordinate is a·(−4) = −4a. So P lies on the line x = −4a — the directrix. Reading backwards, every pair of tangents meeting on the directrix touches the parabola at points whose joining chord subtends 90° at the vertex. One substitution converted a geometric condition into t₁t₂ = −4, and the whole locus fell out of the x-coordinate alone. That is the template: express the configuration's key point in terms of t₁t₂ and t₁ + t₂, translate the geometric condition into an equation on those two symmetric quantities, and eliminate. ## Parameters versus coordinates Main usually hands over a specific point and asks for the tangent or normal — substitute the parameter and the line equation appears. Advanced asks loci, where the parametric route beats coordinate substitution because the symmetry between t₁ and t₂ is preserved and conditions translate into t₁t₂ or t₁ + t₂ cleanly. The traps: quoting t₁t₂ = −1 when the question wants −4 — focal chord and right-angle-at-vertex both appear, and options are built on the confusion; using (a cosθ, b sinθ) on the hyperbola, which traces nothing sensible; and getting the tangent slope wrong as t instead of 1/t — differentiating ty = x + at² with respect to x settles it instantly. A fourth: on xy = c², the tangent is x/t + ty = 2c, and dropping the factor 2 is a silent error that survives every consistency check except substitution of the contact point. ## Frequently asked questions ### When is the chord joining t₁ and t₂ on y² = 4ax a focal chord? Exactly when t₁t₂ = −1; its length a(t + 1/t)² is then minimised at the latus rectum 4a. ### Where do the tangents at t₁ and t₂ meet? At (at₁t₂, a(t₁ + t₂)) — substituting a geometric condition on this point is how most loci are found. ### What is the tangent at the point t on y² = 4ax? ty = x + at², with slope 1/t; the corresponding normal is y = −tx + 2at + at³. ### What condition makes tangents to a parabola meet on the directrix? t₁t₂ = −4, equivalently the chord of contact subtending a right angle at the vertex. ### How is xy = c² parametrised? By (ct, c/t); its tangent at parameter t is x/t + ty = 2c.