Conic Sections for JEE Maths
Conic Sections is one of the most important Coordinate Geometry topics in JEE, typically contributing 2-3 questions per paper. It covers the circle, parabola, ellipse, and hyperbola. The key to mastering conics is knowing the standard forms and properties by heart, and being comfortable with parametric representation.
Prerequisites: Straight Lines, Basic Coordinate Geometry
What JEE Tests in Conic Sections
The most frequently tested areas:
- Standard equations and parametric forms of all four conics
- Tangent and normal equations at a point or in terms of slope
- Condition for a line to be tangent to a conic
- Focal chord properties (especially for parabola)
- Director circle and auxiliary circle (ellipse/hyperbola)
- Eccentricity-based problems
- Locus problems involving conics
The Four Conics at a Glance
Circle: x² + y² = r². Parametric: (rcosθ, rsinθ). Eccentricity = 0.
Parabola: y² = 4ax. Parametric: (at², 2at). Focus (a,0), directrix x = -a. Eccentricity = 1.
Ellipse: x²/a² + y²/b² = 1. Parametric: (acosθ, bsinθ). Eccentricity = √(1 - b²/a²) < 1.
Hyperbola: x²/a² - y²/b² = 1. Parametric: (asecθ, btanθ). Eccentricity = √(1 + b²/a²) > 1.
Tangent Equations — The Most Tested Subtopic
For every conic, you need to know tangent equations in three forms: (1) at a point on the curve, (2) in terms of slope m, (3) using parametric form.
For parabola y² = 4ax: tangent at (at², 2at) is ty = x + at². Tangent with slope m is y = mx + a/m.
Condition for tangency: substitute the line equation into the conic equation and set the discriminant = 0. This technique works universally and is often the fastest approach in JEE.
Key Formulas
Common Mistakes to Avoid
- Mistake: Mixing up a and b in ellipse (a is always the larger semi-axis for standard form)
- Mistake: Forgetting the ± in tangent equations with slope m
- Mistake: Using wrong parametric form for hyperbola (secθ, tanθ not cosθ, sinθ)
- Mistake: Not shifting origin when conic is not centered at origin
How to Practice This Topic
Learn each conic separately: spend 2-3 days on parabola, then ellipse, then hyperbola. For each, master the standard form, parametric form, tangent in all three forms, and normal. Then practice mixed problems.
Formula Sheets
Practice Conic Sections Now
Solve questions with step-by-step explanations on TimeBack
Start Free Practice