JEE Mathematics Conic Sections - Complete PYQ Compilation (2009-2024)

🔮 Overview

This comprehensive compilation covers all aspects of Conic Sections from JEE Mathematics with 15 years of previous year questions (2009-2024). The systematic organization includes circles, parabolas, ellipses, and hyperbolas with detailed analysis and strategic preparation guidance.

📊 Chapter Analysis

Weightage and Distribution

📈 Conic Sections Analysis (2009-2024):

Chapter Weightage: 8-9%
Total Questions: 140+
Average Questions per Year: 8-9
Difficulty Level: Medium to Hard

Question Distribution:
- Circle: 30% (42+ questions)
- Parabola: 25% (35+ questions)
- Ellipse: 25% (35+ questions)
- Hyperbola: 20% (28+ questions)

Year-wise Trend Analysis

📅 Year-wise Question Distribution:

2009-2012 (IIT-JEE Era):
- Total Questions: 48
- Focus: Classical conic properties
- Pattern: Standard applications
- Average Difficulty: Medium-Hard

2013-2016 (JEE Advanced Transition):
- Total Questions: 36
- Focus: Concept-based problems
- Pattern: Mixed conic types
- Average Difficulty: Medium

2017-2020 (Stabilization):
- Total Questions: 32
- Focus: Practical applications
- Pattern: Property-based questions
- Average Difficulty: Medium

2021-2024 (Digital Era):
- Total Questions: 32
- Focus: Complex geometric reasoning
- Pattern: Multi-concept integration
- Average Difficulty: Hard

⭕ Circle

Basic Concepts and Properties

📚 Essential Circle Formulas:

1. Standard Form:
   (x - h)² + (y - k)² = r²
   where (h, k) is center, r is radius

2. General Form:
   x² + y² + 2gx + 2fy + c = 0
   Center: (-g, -f), Radius: √(g² + f² - c)

3. Diameter Form:
   (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0
   where (x₁, y₁) and (x₂, y₂) are endpoints of diameter

4. Special Cases:
   - Center at origin: x² + y² = r²
   - Circle through origin: x² + y² + 2gx + 2fy = 0
   - Tangent condition: S₁ = 0

Previous Year Questions - Circle

📝 Representative Questions:

Example 1 (2023):
Q: Find equation of circle passing through (1, 2), (3, 4), and (5, 2).
Solution: Using general form x² + y² + 2gx + 2fy + c = 0
Substituting points:
1 + 4 + 2g + 4f + c = 0 → 2g + 4f + c = -5
9 + 16 + 6g + 8f + c = 0 → 6g + 8f + c = -25
25 + 4 + 10g + 4f + c = 0 → 10g + 4f + c = -29
Solving: g = -3, f = -2, c = 5
Answer: x² + y² - 6x - 4y + 5 = 0

Example 2 (2022):
Q: Find center and radius of circle x² + y² - 4x + 6y - 12 = 0.
Solution: Complete the squares:
x² - 4x + y² + 6y = 12
(x² - 4x + 4) + (y² + 6y + 9) = 12 + 4 + 9
(x - 2)² + (y + 3)² = 25
Center: (2, -3), Radius: 5

Example 3 (2021):
Q: Find equation of circle touching both axes and passing through (1, 2).
Solution: Let center be (a, a) and radius = a (touches both axes)
Distance from center to (1, 2) = a
√[(1 - a)² + (2 - a)²] = a
(1 - a)² + (2 - a)² = a²
1 - 2a + a² + 4 - 4a + a² = a²
a² - 6a + 5 = 0
a = 1 or a = 5
For a = 1: (x - 1)² + (y - 1)² = 1
For a = 5: (x - 5)² + (y - 5)² = 25

Example 4 (2020):
Q: Find length of common chord of circles x² + y² = 25 and (x - 7)² + y² = 9.
Solution: First circle: center O(0, 0), radius r₁ = 5
Second circle: center C(7, 0), radius r₂ = 3
Distance between centers: d = 7
Common chord length = 2√[r₁² - (d² + r₁² - r₂²)²/(4d²)]
= 2√[25 - (49 + 25 - 9)²/(4×49)]
= 2√[25 - (65)²/(196)]
= 2√[25 - 4225/196]
= 2√[(4900 - 4225)/196]
= 2√(675/196) = 2×15√3/14 = 15√3/7

Tangents and Chords

📐 Tangent and Chord Properties:

1. Tangent at Point (x₁, y₁):
   xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
   For x² + y² = r²: xx₁ + yy₁ = r²

2. Tangent from External Point:
   Length of tangent = √(S₁)
   where S₁ is value obtained by substituting point in circle equation

3. Chord Properties:
   - Perpendicular from center bisects chord
   - Product of segments of intersecting chords
   - Angle between chord and tangent

4. Director Circle:
   x² + y² = 2r² (for x² + y² = r²)
   Contains all points from which tangents are perpendicular

📈 Parabola

Standard Forms and Properties

📚 Essential Parabola Formulas:

1. Right Opening: y² = 4ax
   - Focus: (a, 0)
   - Directrix: x = -a
   - Vertex: (0, 0)
   - Axis: x-axis
   - Latus rectum: 4a

2. Left Opening: y² = -4ax
   - Focus: (-a, 0)
   - Directrix: x = a
   - Vertex: (0, 0)
   - Axis: x-axis
   - Latus rectum: 4a

3. Upward Opening: x² = 4ay
   - Focus: (0, a)
   - Directrix: y = -a
   - Vertex: (0, 0)
   - Axis: y-axis
   - Latus rectum: 4a

4. Downward Opening: x² = -4ay
   - Focus: (0, -a)
   - Directrix: y = a
   - Vertex: (0, 0)
   - Axis: y-axis
   - Latus rectum: 4a

5. Parametric Form: (at², 2at)
6. Focal Property: Distance from focus = Distance from directrix

Previous Year Questions - Parabola

📝 Representative Questions:

Example 1 (2023):
Q: Find focus and directrix of parabola y² = 12x.
Solution: Comparing with y² = 4ax: 4a = 12 → a = 3
Focus: (3, 0), Directrix: x = -3
Latus rectum: 4a = 12

Example 2 (2022):
Q: Find equation of parabola with focus (2, 3) and directrix x = 0.
Solution: Let point (x, y) be on parabola
Distance from focus = Distance from directrix
√[(x - 2)² + (y - 3)²] = |x - 0|
Squaring: (x - 2)² + (y - 3)² = x²
x² - 4x + 4 + y² - 6y + 9 = x²
y² - 6y - 4x + 13 = 0

Example 3 (2021):
Q: Find vertex of parabola y² - 4y - 8x + 12 = 0.
Solution: Complete the square for y:
y² - 4y = 8x - 12
y² - 4y + 4 = 8x - 12 + 4
(y - 2)² = 8x - 8
(y - 2)² = 8(x - 1)
Vertex: (1, 2)

Example 4 (2020):
Q: Find length of latus rectum of parabola (y - 2)² = 8(x - 3).
Solution: Comparing with (y - k)² = 4a(x - h)
4a = 8 → a = 2
Length of latus rectum = 4a = 8

Example 5 (2019):
Q: Find equation of tangent to parabola y² = 4x at point (1, 2).
Solution: Using tangent formula: yy₁ = 2(x + x₁)
y × 2 = 2(x + 1)
2y = 2x + 2
y = x + 1

Example 6 (2018):
Q: Find equation of parabola with vertex at (2, 1), axis parallel to x-axis, and passing through (4, 3).
Solution: Equation: (y - 1)² = 4a(x - 2)
Substituting (4, 3): (3 - 1)² = 4a(4 - 2)
4 = 8a → a = 1/2
Final equation: (y - 1)² = 2(x - 2)

Advanced Parabola Properties

🔍 Advanced Parabola Concepts:

1. Tangent in Parametric Form:
   ty = x + at² (for y² = 4ax)
   where point is (at², 2at)

2. Normal in Parametric Form:
   y = -tx + 2at + at³

3. Focal Chord:
   If endpoints are (at₁², 2at₁) and (at₂², 2at₂)
   Then t₁t₂ = -1

4. Chord of Contact:
   T = 0 from external point (x₁, y₁)

5. Director Circle:
   x² + y² = 2a² + 2ah (for shifted parabola)

🥚 Ellipse

Standard Forms and Properties

📚 Essential Ellipse Formulas:

1. Horizontal Major Axis: x²/a² + y²/b² = 1 (a > b)
   - Center: (0, 0)
   - Major axis: 2a (along x-axis)
   - Minor axis: 2b (along y-axis)
   - Vertices: (±a, 0)
   - Foci: (±c, 0) where c² = a² - b²
   - Eccentricity: e = c/a
   - Directrices: x = ±a/e

2. Vertical Major Axis: x²/b² + y²/a² = 1 (a > b)
   - Center: (0, 0)
   - Major axis: 2a (along y-axis)
   - Minor axis: 2b (along x-axis)
   - Vertices: (0, ±a)
   - Foci: (0, ±c) where c² = a² - b²
   - Eccentricity: e = c/a
   - Directrices: y = ±a/e

3. Special Properties:
   - Sum of distances from foci = 2a
   - Area = πab
   - Perimeter ≈ π[3(a + b) - √{(3a + b)(a + 3b)}]

Previous Year Questions - Ellipse

📝 Representative Questions:

Example 1 (2023):
Q: Find eccentricity of ellipse 9x² + 16y² = 144.
Solution: Standard form: x²/16 + y²/9 = 1
Here a² = 16, b² = 9, a = 4, b = 3
c² = a² - b² = 16 - 9 = 7
e = c/a = √7/4

Example 2 (2022):
Q: Find equation of ellipse with foci (±3, 0) and major axis 10.
Solution: c = 3, 2a = 10 → a = 5
c² = a² - b² → 9 = 25 - b² → b² = 16
Equation: x²/25 + y²/16 = 1

Example 3 (2021):
Q: Find equation of tangent to ellipse x²/25 + y²/16 = 1 at point (3, 8/5).
Solution: Using tangent formula: xx₁/a² + yy₁/b² = 1
3x/25 + (8y/5)/16 = 1
3x/25 + 8y/80 = 1
3x/25 + y/10 = 1
6x + 5y = 50

Example 4 (2020):
Q: Find area of ellipse 4x² + 9y² = 36.
Solution: Standard form: x²/9 + y²/4 = 1
a² = 9 → a = 3, b² = 4 → b = 2
Area = πab = π × 3 × 2 = 6π

Example 5 (2019):
Q: Find equation of ellipse with center (2, 1), horizontal major axis of length 10, and passing through (5, 4).
Solution: Center (h, k) = (2, 1)
Equation: (x - 2)²/a² + (y - 1)²/b² = 1
2a = 10 → a = 5
Substituting (5, 4): (5 - 2)²/25 + (4 - 1)²/b² = 1
9/25 + 9/b² = 1 → 9/b² = 16/25 → b² = 225/16
Final equation: (x - 2)²/25 + (y - 1)²/(225/16) = 1

Advanced Ellipse Properties

🔍 Advanced Ellipse Concepts:

1. Parametric Form:
   x = a cos θ, y = b sin θ
   where 0 ≤ θ < 2π

2. Tangent in Parametric Form:
   x cos θ/a + y sin θ/b = 1

3. Normal in Parametric Form:
   ax/ cos θ - by/ sin θ = a² - b²

4. Director Circle:
   x² + y² = a² + b²

5. Auxiliary Circle:
   x² + y² = a²

6. Eccentric Angle:
   Angle made by radius vector with positive x-axis

📐 Hyperbola

Standard Forms and Properties

📚 Essential Hyperbola Formulas:

1. Horizontal Transverse Axis: x²/a² - y²/b² = 1
   - Center: (0, 0)
   - Transverse axis: 2a (along x-axis)
   - Conjugate axis: 2b (along y-axis)
   - Vertices: (±a, 0)
   - Foci: (±c, 0) where c² = a² + b²
   - Eccentricity: e = c/a
   - Asymptotes: y = ±(b/a)x

2. Vertical Transverse Axis: y²/a² - x²/b² = 1
   - Center: (0, 0)
   - Transverse axis: 2a (along y-axis)
   - Conjugate axis: 2b (along x-axis)
   - Vertices: (0, ±a)
   - Foci: (0, ±c) where c² = a² + b²
   - Eccentricity: e = c/a
   - Asymptotes: y = ±(a/b)x

3. Rectangular Hyperbola:
   When a = b: xy = c²
   Asymptotes are perpendicular
   Eccentricity = √2

4. Special Properties:
   - Difference of distances from foci = 2a
   - Product of distances from asymptotes = constant

Previous Year Questions - Hyperbola

📝 Representative Questions:

Example 1 (2023):
Q: Find asymptotes of hyperbola 16x² - 9y² = 144.
Solution: Standard form: x²/9 - y²/16 = 1
a² = 9, b² = 16, a = 3, b = 4
Asymptotes: y = ±(b/a)x = ±(4/3)x

Example 2 (2022):
Q: Find eccentricity of hyperbola 9x² - 16y² = 144.
Solution: Standard form: x²/16 - y²/9 = 1
a² = 16, b² = 9, a = 4, b = 3
c² = a² + b² = 16 + 9 = 25 → c = 5
e = c/a = 5/4

Example 3 (2021):
Q: Find equation of hyperbola with foci (±5, 0) and vertices (±3, 0).
Solution: c = 5, a = 3
c² = a² + b² → 25 = 9 + b² → b² = 16
Equation: x²/9 - y²/16 = 1

Example 4 (2020):
Q: Find equation of tangent to hyperbola x²/9 - y²/4 = 1 at point (6, 4).
Solution: Using tangent formula: xx₁/a² - yy₁/b² = 1
6x/9 - 4y/4 = 1
2x/3 - y = 1
2x - 3y = 3

Example 5 (2019):
Q: Find equation of rectangular hyperbola with asymptotes x = 0 and y = 0, passing through (2, 3).
Solution: For rectangular hyperbola with coordinate axes as asymptotes: xy = c²
Substituting (2, 3): 2 × 3 = c² → c² = 6
Equation: xy = 6

Example 6 (2018):
Q: Find equation of hyperbola with center (2, 1), transverse axis parallel to x-axis, vertices (5, 1) and (-1, 1).
Solution: Center (2, 1), vertices at distance a = 3 from center
Equation: (x - 2)²/9 - (y - 1)²/b² = 1
Need more information to find b²

Advanced Hyperbola Properties

🔍 Advanced Hyperbola Concepts:

1. Parametric Form:
   x = a sec θ, y = b tan θ
   where θ is parameter

2. Tangent in Parametric Form:
   x sec θ/a - y tan θ/b = 1

3. Normal in Parametric Form:
   ax/ sec θ + by/ tan θ = a² + b²

4. Director Circle:
   x² + y² = a² - b² (if a > b)

5. Auxiliary Circle:
   x² + y² = a²

6. Conjugate Hyperbola:
   -y²/b² + x²/a² = 1
   Shares same asymptotes

📊 Performance Analysis

Difficulty Level Distribution

📊 Conic Sections Difficulty Analysis (2009-2024):

Circle (30% of questions):
- Easy: 50% (Basic equations and properties)
- Medium: 35% (Tangents and chords)
- Hard: 15% (Complex circle problems)

Parabola (25% of questions):
- Easy: 35% (Standard forms and basic properties)
- Medium: 45% (Tangents and applications)
- Hard: 20% (Advanced parabola problems)

Ellipse (25% of questions):
- Easy: 30% (Standard forms and basic properties)
- Medium: 50% (Applications and tangents)
- Hard: 20% (Complex ellipse problems)

Hyperbola (20% of questions):
- Easy: 25% (Standard forms and basic properties)
- Medium: 45% (Applications and asymptotes)
- Hard: 30% (Complex hyperbola problems)

Success Rate by Topic

📯 Student Performance Analysis:

High Success (>70%):
- Basic circle equations
- Standard parabola forms
- Basic ellipse properties
- Standard hyperbola equations

Medium Success (50-70%):
- Tangents to conics
- Focus-directrix properties
- Latus rectum calculations
- Asymptotes of hyperbola

Low Success (<50%):
- Complex conic applications
- Parametric forms
- Advanced properties
- Mixed conic problems

🎯 Strategic Preparation

Study Priority Matrix

🎯 Topic Priority Ranking:

High Priority (Must Master):
1. Circle (30% weightage)
   - Standard and general forms
   - Center and radius calculations
   - Tangents and chords
   - Applications

2. Parabola (25% weightage)
   - Standard forms
   - Focus and directrix
   - Tangents and normals
   - Parametric forms

Medium Priority (Important):
3. Ellipse (25% weightage)
   - Standard forms
   - Eccentricity and foci
   - Tangents and properties
   - Applications

4. Hyperbola (20% weightage)
   - Standard forms
   - Asymptotes and foci
   - Eccentricity
   - Basic applications

Problem-Solving Strategy

🧠 Conic Sections Problem-Solving Approach:

1. Identify the Conic Type:
   - Look at the equation form
   - Check coefficients and signs
   - Identify standard form parameters

2. Extract Key Information:
   - Find center, focus, vertices
   - Calculate eccentricity
   - Identify special properties

3. Choose Solution Method:
   - Direct formula application
   - Parametric approach
   - Geometric properties
   - Coordinate transformations

4. Verify the Solution:
   - Check if satisfies given conditions
   - Verify special cases
   - Cross-validate with properties

Common Mistakes to Avoid

⚠️ Common Errors in Conic Sections:

1. Identification Errors:
   - Wrong conic type identification
   - Confusing ellipse and hyperbola
   - Incorrect parameter extraction

2. Formula Errors:
   - Wrong focus/directrix formulas
   - Incorrect eccentricity calculation
   - Wrong tangent equations

3. Calculation Errors:
   - Sign mistakes in completing squares
   - Arithmetic errors in distances
   - Incorrect substitution

4. Special Cases:
   - Missing degenerate cases
   - Incorrect handling of shifted conics
   - Wrong asymptote calculations

📝 Practice Questions

Circle Practice

📚 Circle Practice Questions:

Easy Level:
1. Find center and radius of x² + y² - 6x + 4y - 12 = 0
2. Find equation of circle with center (2, -3) and radius 5
3. Find equation of circle through (1, 2) touching both axes

Medium Level:
4. Find equation of circle passing through (0, 0), (2, 0), (0, 2)
5. Find length of tangent from (5, 3) to x² + y² = 25
6. Find equation of circle touching x = 0, y = 0, and x + y = 2

Hard Level:
7. Find radical axis of circles x² + y² = 25 and (x - 4)² + (y - 3)² = 9
8. Find equation of circle passing through (1, 2) and orthogonal to x² + y² = 4
9. Find locus of center of circle touching x-axis and passing through (2, 3)

Solutions:
1. Complete squares: (x - 3)² + (y + 2)² = 25
   Center: (3, -2), Radius: 5

2. (x - 2)² + (y + 3)² = 25

3. Let center be (a, a), radius = a
   (1 - a)² + (2 - a)² = a²
   a² - 3a + 5 = 0 → No real solution
   Try center (a, -a): (1 - a)² + (2 + a)² = a² → a = 5
   (x - 5)² + (y + 5)² = 25

Parabola Practice

📚 Parabola Practice Questions:

Easy Level:
1. Find focus and directrix of y² = 16x
2. Find equation of parabola with focus (0, -2) and directrix y = 2
3. Find vertex of parabola x² - 4x - 8y + 12 = 0

Medium Level:
4. Find equation of tangent to y² = 4x at (1, 2)
5. Find equation of parabola with vertex (2, 3) and focus (5, 3)
6. Find length of latus rectum of (y - 2)² = 12(x - 1)

Hard Level:
7. Find equation of parabola passing through (0, 0), (1, 2), (2, 6)
8. Find equation of tangent to y² = 8x which makes angle 45° with x-axis
9. Find focal chord of y² = 4ax with slope m

Solutions:
1. 4a = 16 → a = 4
   Focus: (4, 0), Directrix: x = -4

2. Let (x, y) be point on parabola
   √[(x - 0)² + (y + 2)²] = |y - 2|
   x² + (y + 2)² = (y - 2)²
   x² + y² + 4y + 4 = y² - 4y + 4
   x² + 8y = 0

Ellipse Practice

📚 Ellipse Practice Questions:

Easy Level:
1. Find eccentricity of 4x² + 9y² = 36
2. Find equation of ellipse with foci (±2, 0) and major axis 8
3. Find area of ellipse x²/25 + y²/9 = 1

Medium Level:
4. Find equation of tangent to x²/16 + y²/9 = 1 at (4, 0)
5. Find equation of ellipse with center (1, 2), horizontal major axis 10, passing through (4, 4)
6. Find eccentricity if sum of focal distances is 10 and distance between foci is 8

Hard Level:
7. Find equation of ellipse with eccentricity 1/2 and focal distance 6
8. Find locus of center of ellipse touching coordinate axes
9. Find equation of ellipse passing through (0, 3), (4, 0), (0, -3)

Solutions:
1. Standard form: x²/9 + y²/4 = 1
   a² = 9, b² = 4, c² = a² - b² = 5
   e = √5/3

2. c = 2, 2a = 8 → a = 4
   c² = a² - b² → 4 = 16 - b² → b² = 12
   x²/16 + y²/12 = 1

3. a² = 25, b² = 9
   Area = πab = π × 5 × 3 = 15π

Hyperbola Practice

📚 Hyperbola Practice Questions:

Easy Level:
1. Find asymptotes of 4x² - 9y² = 36
2. Find eccentricity of x²/9 - y²/16 = 1
3. Find equation of hyperbola with vertices (±3, 0) and foci (±5, 0)

Medium Level:
4. Find equation of tangent to x²/9 - y²/4 = 1 at (6, 4)
5. Find equation of rectangular hyperbola with asymptotes x + y = 0 and x - y = 0
6. Find eccentricity if distance between foci is 10 and between vertices is 6

Hard Level:
7. Find equation of hyperbola with eccentricity 2 and focal distance 8
8. Find equation of hyperbola conjugate to x²/9 - y²/4 = 1
9. Find locus of foot of perpendicular from focus to any tangent of hyperbola

Solutions:
1. Standard form: x²/9 - y²/4 = 1
   a = 3, b = 2
   Asymptotes: y = ±(2/3)x

2. a² = 9, b² = 4, c² = a² + b² = 13
   e = √13/3

3. a = 3, c = 5
   c² = a² + b² → 25 = 9 + b² → b² = 16
   x²/9 - y²/16 = 1

🏆 Success Tips

High-Scoring Strategies

🎯 Tips for Maximizing Scores in Conic Sections:

1. Master Standard Forms:
   - Practice identifying conic types quickly
   - Memorize all standard forms
   - Understand parameter relationships

2. Focus on Properties:
   - Learn focus-directrix relationships
   - Master eccentricity calculations
   - Understand special properties

3. Practice Tangents:
   - Learn all tangent forms
   - Practice point of tangency problems
   - Understand chord properties

4. Visualization Skills:
   - Develop geometric intuition
   - Practice sketching conics
   - Understand parameter effects

Time Management

⏱️ Time Allocation for Conic Sections Questions:

Easy Questions: 3-4 minutes each
- Basic form identification
- Simple property applications
- Standard parameter calculations

Medium Questions: 5-7 minutes each
- Tangent and chord problems
- Mixed conic applications
- Property-based questions

Hard Questions: 8-12 minutes each
- Complex geometric relationships
- Parametric applications
- Advanced properties

Strategy:
- Identify conic type quickly
- Choose appropriate solution method
- Work systematically through calculations

🎓 Conclusion

Conic Sections is a significant topic in coordinate geometry that tests both conceptual understanding and problem-solving skills. With systematic practice of all conic types and their properties, students can excel in this area.

Key Takeaways

✅ Master all standard forms of conics
✅ Understand focus-directrix properties
✅ Practice tangent and chord problems
✅ Develop geometric visualization
✅ Apply properties to solve complex problems

Final Advice

🎯 Success in Conic Sections requires:
- Strong foundation in all conic types
- Quick identification of conic properties
- Practice with various problem types
- Good visualization skills
- Systematic approach to problem-solving

Remember: Each conic type has its own properties and applications. Master them individually and then practice mixed problems for comprehensive understanding! 🔮

Master Conic Sections with systematic preparation and comprehensive practice of 15 years of JEE previous year questions! 🔮

With dedicated practice and clear understanding of conic properties, this topic can become a high-scoring area in JEE Mathematics! 🎯