Last Updated: 11 May 2026
JEE Main 2027 Mathematics carries 5-6 questions worth 20-24 marks from Coordinate Geometry — second only to Calculus. Conic Sections (circles + 3 conics) account for 70% of that block.
Circle — Standard Forms
- General: x² + y² + 2gx + 2fy + c = 0 → centre (-g, -f), radius √(g²+f²-c).
- Standard: (x-h)² + (y-k)² = r².
- Parametric: x = h + r cosθ, y = k + r sinθ.
- Equation of tangent at (x₁,y₁): xx₁ + yy₁ + g(x+x₁) + f(y+y₁) + c = 0.
- Power of a point: x₁² + y₁² + 2gx₁ + 2fy₁ + c (positive → outside, negative → inside).
Parabola — y² = 4ax
| Vertex | (0, 0) |
| Focus | (a, 0) |
| Directrix | x = -a |
| Axis | x-axis |
| Latus Rectum | 4a |
| Parametric | (at², 2at) |
| Eccentricity | e = 1 |
Tangent at (at², 2at): ty = x + at². Normal: y + tx = 2at + at³.
Ellipse — x²/a² + y²/b² = 1 (a > b)
| Vertices | (±a, 0) |
| Foci | (±ae, 0) |
| Directrix | x = ±a/e |
| Eccentricity | e = √(1 – b²/a²) |
| Latus Rectum | 2b²/a |
| SP + S’P | 2a (focal chord property) |
Tangent at (a cosθ, b sinθ): (x/a)cosθ + (y/b)sinθ = 1.
Hyperbola — x²/a² – y²/b² = 1
| Vertices | (±a, 0) |
| Foci | (±ae, 0) |
| Eccentricity | e = √(1 + b²/a²) |
| Latus Rectum | 2b²/a |
| Asymptotes | y = ±(b/a)x |
| |SP – S’P| | 2a |
| Conjugate hyperbola | y²/b² – x²/a² = 1 |
Director Circle
- Circle: locus of pairs of perpendicular tangents — concentric circle of radius r√2.
- Parabola: directrix itself.
- Ellipse: x² + y² = a² + b².
- Hyperbola: x² + y² = a² – b².
Common Properties Across Conics
- Reflection property: parabola — parallel rays focus; ellipse — ray from one focus reflects to the other; hyperbola — appears to come from other focus.
- Focal chord through (at², 2at) has the other end at (a/t², -2a/t).
- Latus rectum length: parabola 4a, ellipse/hyperbola 2b²/a.
- Eccentricity comparison: e_circle = 0, e_ellipse < 1, e_parabola = 1, e_hyperbola > 1.
JEE Main 2026 Coordinate Geometry Snapshot
- Q1: Tangent to circle from external point — slope form
- Q2: Locus of midpoint of chord in parabola
- Q3: Foci distance product on ellipse tangent
- Q4: Asymptote angle of rectangular hyperbola
- Q5: Common chord of two circles
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FAQ
How many marks from Coordinate Geometry in JEE Main?
20-24 marks (5-6 questions).
Most-asked conic?
Parabola — slightly higher frequency due to JEE Main’s preference for focal chord properties.
Is JEE Advanced harder here?
Significantly. Advanced often combines two conics or uses parametric integration; Main keeps it standard.
Practice Quiz
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