JEE Main Maths · Conic Sections PYQ
JEE Main Conic Sections Previous Year Questions (2002–2025)
Conic Sections is the largest-yield chapter in Coordinate Geometry — ~6% weightage across 10 years with consistent 1–2 questions per JEE Main session. Parabola, ellipse and hyperbola share a single algebraic framework with predictable parametric tricks, making this a high-ROI P1 chapter once you lock in the standard-form results.
Conic Sections PYQs from 2002 to 2025, covering parabola (standard form, focus-directrix, tangent), ellipse (auxiliary circle, eccentricity, parametric), hyperbola (asymptotes, director circle) and cross-conic problems. Every solution uses KaTeX and cites the parametric shortcut where available.
Conic Sections (Parabola, Ellipse, Hyperbola) at a Glance
Key Sub-Topics & What's Tested
Parabola — Standard Form & Properties
y² = 4ax and four standard orientations, focus, directrix, vertex, latus rectum, focal chord properties.
Parabola — Tangent, Normal, Chord
Tangent at (at², 2at), slope form of tangent, chord of contact from external point, chord with mid-point.
Ellipse — Standard Form & Eccentricity
x²/a² + y²/b² = 1, relationship b² = a²(1 - e²), major/minor axes, auxiliary circle, parametric (a cos θ, b sin θ).
Ellipse — Tangent & Normal
Tangent at parametric point (a cos θ, b sin θ), slope form, director circle (x² + y² = a² + b²).
Hyperbola — Standard Form & Asymptotes
x²/a² - y²/b² = 1, eccentricity e² = 1 + b²/a², asymptotes y = ±(b/a)x, conjugate hyperbola, rectangular hyperbola.
Hyperbola — Tangent & Normal
Parametric form (a sec θ, b tan θ), tangent and normal equations, director circle (x² + y² = a² - b²) — only if a > b.
Focal Chord & Properties
Semi-latus rectum relation, harmonic mean of focal distances, length of focal chord in parabola, focal chord in ellipse/hyperbola.
Locus Problems
Finding locus of mid-point of chord, foot of perpendicular from focus, intersection of tangents — algebraic elimination with conic parameters.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Standard Form Manipulation | 20% | Given a parabola in general form, find focus and directrix. |
| Tangent & Normal Equations | 25% | Find equation of tangent to x²/9 + y²/4 = 1 parallel to 3x + 2y = 7. |
| Chord of Contact / Mid-Point | 15% | Find chord of contact from external point (3, 4) to parabola y² = 8x. |
| Locus / Parametric Problems | 15% | Find locus of mid-point of chord of y² = 4ax passing through focus. |
| Asymptote / Director Circle | 10% | Find director circle of ellipse x²/16 + y²/9 = 1. |
| Cross-Conic / Combined | 15% | A chord of a parabola also lies on an ellipse — find common property. |
How to Solve Conic Sections (Parabola, Ellipse, Hyperbola) PYQs
- 1Lock the parametric forms. Parabola: (at², 2at). Ellipse: (a cos θ, b sin θ). Hyperbola: (a sec θ, b tan θ). Almost every PYQ reduces to algebra in θ or t.
- 2For tangent problems, use slope form first. y = mx ± √(a²m² - b²) for ellipse; y = mx + a/m for parabola. Avoids point-slope derivation.
- 3Chord of contact template. From external point (x₁, y₁) to conic S = 0, chord of contact is T = 0 where T substitutes x·x₁ for x², y·y₁ for y² etc. Fixed template.
- 4Locus problems: parameterise first. Express the quantity in terms of conic parameter (t for parabola, θ for ellipse/hyperbola), then eliminate the parameter using algebraic identity.
- 5Director circle is rapid recall. Ellipse: x² + y² = a² + b². Hyperbola: x² + y² = a² - b² (real only if a > b). Parabola: no director circle. Know these three.
Common Mistakes That Cost Marks
- Wrong sign in hyperbola formula. b² = a²(e² - 1) for hyperbola, not a²(1 - e²) (that's ellipse). Mixing these breaks every eccentricity calculation.
- Confusing parametric forms. Ellipse uses sin/cos (bounded), hyperbola uses sec/tan (unbounded). Using sin/cos for hyperbola gives wrong results.
- Missing the ± in slope-form tangent. For ellipse, y = mx ± √(a²m² + b²) gives two tangents. Missing one loses half the answer.
- Wrong focal distance for ellipse. Sum of focal distances = 2a (not 2a·e or a + b). This is the defining property of an ellipse.
- Treating parabola like ellipse. Parabola has one focus and one directrix; no auxiliary circle, no second focus. Don't borrow ellipse properties for parabola problems.
Related JEE Main Practice
Frequently asked questions
How many Conic Sections PYQs should I solve?
Target 70–90 PYQs across 2010–2025. Given the high template repeat rate (parametric tricks appear in nearly every PYQ), this volume builds strong pattern recognition.
Which conic has the most JEE Main questions?
Parabola — roughly 40% of Conic Sections PYQs. Ellipse ~35%, Hyperbola ~25%. All three follow similar template logic, so mastering parabola first makes the others easier.
Do I need to memorise the auxiliary circle concept?
Yes — it's frequently tested. Auxiliary circle of ellipse x²/a² + y²/b² = 1 is x² + y² = a². Parametric point (a cos θ, b sin θ) on ellipse corresponds to (a cos θ, a sin θ) on auxiliary circle. Critical for several PYQ templates.
How do rectangular hyperbola PYQs differ?
Rectangular hyperbola has perpendicular asymptotes — a = b in the standard form. Equation becomes xy = c². Different parametric form (ct, c/t). Comes up in 2–3 PYQs per 10 years as a distinct template.
Is Parabola also in JEE Advanced syllabus?
Yes — Conic Sections is heavily tested in JEE Advanced too, often as multi-concept problems with Calculus or 3D Geometry. Your Main-level PYQ mastery transfers directly, but add multi-step problems for Advanced.
Do I need 3D versions of conics?
No — JEE Main Conic Sections is strictly 2D. 3D geometry is a separate chapter with its own PYQs (lines in 3D, planes, direction cosines).
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