Last Updated: April 2026
Coordinate Geometry is one of the most high-yielding units in JEE Main Mathematics, contributing 4-7 questions every year across Straight Lines, Circles, and Conic Sections. The Conic Sections chapter alone (Parabola, Ellipse, Hyperbola) can fetch you 8-12 marks in JEE Main 2027. This complete guide covers all key formulas, properties, and 40 practice MCQs.
JEE Main Coordinate Geometry — Chapter-wise Weightage
| Topic | JEE Main Questions/Year (Avg) | Marks | Difficulty |
|---|---|---|---|
| Straight Lines | 1-2 | 4-8 | Easy-Medium |
| Circle | 1-2 | 4-8 | Medium |
| Parabola | 1-2 | 4-8 | Medium-Hard |
| Ellipse | 1-2 | 4-8 | Medium-Hard |
| Hyperbola | 1 | 4 | Hard |
| Total | 5-9 | 20-36 | — |
Section 1: Circle
Standard Forms
- Standard form: x² + y² = r² (centre at origin, radius r)
- General form: x² + y² + 2gx + 2fy + c = 0 (centre: (-g, -f), radius: √(g²+f²-c))
- Diameter form: (x-x₁)(x-x₂) + (y-y₁)(y-y₂) = 0
Key Results for Circles
| Concept | Formula |
|---|---|
| Length of tangent from point (h,k) | √(h²+k²+2gh+2fk+c) |
| Equation of tangent at point (x₁,y₁) on x²+y²=r² | xx₁ + yy₁ = r² |
| Condition for line y=mx+c to be tangent to x²+y²=r² | c² = r²(1+m²) |
| Angle between two circles (angle of intersection) | cos θ = (r₁²+r₂²-d²)/(2r₁r₂) |
| Condition for orthogonal circles | 2g₁g₂ + 2f₁f₂ = c₁+c₂ |
Section 2: Parabola
Standard Equations and Key Elements
| Form | Vertex | Focus | Directrix | Axis | LR Length |
|---|---|---|---|---|---|
| y² = 4ax (a>0) | (0,0) | (a,0) | x = -a | x-axis | 4a |
| y² = -4ax (a>0) | (0,0) | (-a,0) | x = a | x-axis | 4a |
| x² = 4ay (a>0) | (0,0) | (0,a) | y = -a | y-axis | 4a |
| x² = -4ay (a>0) | (0,0) | (0,-a) | y = a | y-axis | 4a |
Parabola — Key Formulas
- Parametric form: (at², 2at) for y² = 4ax
- Equation of tangent at (at², 2at): ty = x + at²
- Equation of normal at (at², 2at): y + tx = 2at + at³
- Condition for tangency of y=mx+c: c = a/m (for y²=4ax)
- Length of chord through focus: 4a/sin²θ (where θ is angle with axis)
Section 3: Ellipse
Standard Equation: x²/a² + y²/b² = 1 (a > b)
| Element | Formula (a > b) |
|---|---|
| Centre | (0, 0) |
| Vertices | (±a, 0) major; (0, ±b) minor |
| Foci | (±ae, 0) where e = eccentricity |
| Eccentricity e | e = c/a, where c² = a²-b²; 0 < e < 1 |
| Directrices | x = ±a/e |
| Length of major axis | 2a |
| Length of minor axis | 2b |
| Length of latus rectum | 2b²/a |
| Tangent at (x₁,y₁) | xx₁/a² + yy₁/b² = 1 |
Section 4: Hyperbola
Standard Equation: x²/a² – y²/b² = 1
| Element | Formula |
|---|---|
| Eccentricity | e = c/a, c² = a²+b²; e > 1 |
| Foci | (±ae, 0) |
| Vertices | (±a, 0) |
| Asymptotes | y = ±(b/a)x |
| Conjugate hyperbola | -x²/a² + y²/b² = 1 |
| Rectangular hyperbola | xy = c² (when a = b) |
| Length of latus rectum | 2b²/a |
Practice MCQs — JEE Coordinate Geometry 2027
Q1. The length of the tangent drawn from the point (2, 3) to the circle x² + y² – 6x – 4y + 12 = 0 is:
(A) 1 (B) √2 (C) √3 (D) 2
Answer: (A) Length = √(4+9-12-12+12) = √1 = 1
Q2. The eccentricity of the ellipse x²/25 + y²/16 = 1 is:
(A) 3/5 (B) 4/5 (C) 3/4 (D) 5/3
Answer: (A) a²=25, b²=16; c²=25-16=9; e=c/a=3/5
Q3. The focus of the parabola y² = -16x is at:
(A) (4, 0) (B) (-4, 0) (C) (0, 4) (D) (0, -4)
Answer: (B) y²=-4ax → 4a=16 → a=4; focus at (-a, 0) = (-4, 0)
Q4. The asymptotes of the hyperbola x²/9 – y²/16 = 1 are:
(A) y = ±(3/4)x (B) y = ±(4/3)x (C) y = ±(9/16)x (D) y = ±(16/9)x
Answer: (B) Asymptotes: y = ±(b/a)x = ±(4/3)x
Q5. If a circle passes through (0,0), (a,0), (0,b), its centre is:
(A) (a,b) (B) (a/2, b/2) (C) (a/2, 0) (D) (0, b/2)
Answer: (B) The centre of the circumscribed circle of a right triangle (right angle at origin) is the midpoint of the hypotenuse = (a/2, b/2)
Frequently Asked Questions
How many questions from Coordinate Geometry appear in JEE Main 2027?
JEE Main 2027 typically has 5-9 questions from Coordinate Geometry (Straight Lines + Circles + Conic Sections combined). In recent years (2022-2025), JEE Main has had 2-3 questions from Conic Sections alone, making it the third-highest weightage unit in Mathematics after Calculus and Algebra. Expect 20-36 marks from Coordinate Geometry in both Session 1 and Session 2 of JEE Main.
Which Conic Section is most important for JEE Main — Parabola, Ellipse, or Hyperbola?
Parabola and Ellipse each typically contribute 1-2 questions per JEE Main session, making them the most important conic sections. Hyperbola is slightly less frequent (about 1 question per session). For JEE Advanced, all three are equally important and problems are significantly harder. Master Parabola first (simpler properties), then Ellipse (most formula-heavy), then Hyperbola (asymptotes and conjugate hyperbola).
Is Coordinate Geometry scoring in JEE Main?
Yes, Coordinate Geometry is one of the most scoring units in JEE Main Mathematics. The questions follow predictable patterns — most involve finding tangents, normals, locus problems, or chord of contact. Unlike Calculus where computation errors are common, Coordinate Geometry questions can often be solved by careful substitution. Regular formula revision + 30 minutes of daily practice can secure 20+ marks from this unit.
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