Coordinate Geometry & Conic Sections
Practice JEE Main Circles, Parabola, Ellipse & Hyperbola PYQs online. Master tangency condition c = a/m, director circle, eccentricity & step-by-step solutions.
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🎯 High-Yield Syllabus Breakdown
These high-frequency concepts account for over 90% of all questions asked in Coordinate Geometry & Conic Sections:
- Straight Lines & Family of Lines
- Circles: Tangents, Normals & Orthogonality Condition
- Parabola: Standard Form y² = 4ax & Tangent y = mx + a/m
- Ellipse: Standard Form x²/a² + y²/b² = 1, Eccentricity e = √(1 - b²/a²)
- Hyperbola: Standard Form x²/a² - y²/b² = 1 & Rectangular Hyperbola xy = c²
⚡ Essential Formulas & Speed Shortcuts
Formulas and time-saving shortcuts to solve Conic Sections questions in under 45 seconds:
Parabola Tangent Condition: c = a / m for y = mx + c.
Ellipse Tangent Condition: c² = a² m² + b².
Hyperbola Tangent Condition: c² = a² m² - b².
Director Circle (Locus of perpendicular tangents): x² + y² = a² + b² (Ellipse), x² + y² = a² - b² (Hyperbola).
📖 Sample Solved Exam Questions
Preview authentic questions from this chapter with full step-by-step solutions:
💡 View Step-by-Step Solution & Answer
Correct Answer: 8
- Step 1: Complete the square in x: x² - 4x = (x - 2)² - 4.
- Step 2: Rewrite the equation: (x - 2)² - 4 - 8y + 12 = 0 ⇒ (x - 2)² = 8y - 8.
- Step 3: Factor out the coefficient of y: (x - 2)² = 8(y - 1).
- Step 4: Compare with the standard form (X)² = 4a(Y): here 4a = 8. The length of the latus rectum is 4a = 8.
⚡ Speed Tip: Latus Rectum of Parabola: Always equals the coefficient of the linear variable (8).
💡 View Step-by-Step Solution & Answer
Correct Answer: 1/√2
- Step 1: Length of major axis = 2a. Length of latus rectum = 2b²/a.
- Step 2: Given 2b²/a = ½ (2a) = a ⇒ 2b² = a² ⇒ b²/a² = 1/2.
- Step 3: Use the eccentricity formula for an ellipse: e² = 1 - b²/a².
- Step 4: Substitute b²/a² = 1/2: e² = 1 - 1/2 = 1/2 ⇒ e = 1/√2.
⚡ Speed Tip: Direct Eccentricity: 2b²/a = a ⇒ b²/a² = 1/2. e = √(1 - 1/2) = 1/√2.
💡 View Step-by-Step Solution & Answer
Correct Answer: 1
- Step 1: The condition for the line y = mx + c to be tangent to the parabola y² = 4ax is c = a / m.
- Step 2: For y² = 4x, 4a = 4 ⇒ a = 1.
- Step 3: From the line y = mx + 1, the y-intercept is c = 1.
- Step 4: Substitute into condition: 1 = 1 / m ⇒ m = 1.
⚡ Speed Tip: Tangency Condition: For y² = 4ax and y = mx + c, c = a/m. Here 1 = 1/m ⇒ m = 1.
❓ Frequently Asked Questions (FAQs)
What is the condition of tangency for parabola y² = 4ax?
A straight line y = mx + c touches the parabola y² = 4ax if and only if c = a / m.