JEE Mathematics 3D Geometry Sphere Equation Intersection with PlaneLine

What This Is and Why It Matters for JEE

3D Geometry: Sphere is a fundamental topic in JEE, appearing in 2-3 questions every year. It's a moderate difficulty topic, with a slight emphasis on Advanced. Understanding the equation, intersection with plane/line, and key properties is crucial for solving problems accurately.

Prerequisites

• Coordinate Geometry: Understand the basics of coordinate systems, distance formula, and equations of lines and planes.
• Mathematical Reasoning: Familiarize yourself with mathematical induction, proof by contradiction, and logical deductions.
• Geometry: Know basic concepts of points, lines, planes, and solids, including properties of congruent and similar figures.

Core Concepts (Exam-Focused)

• Equation of a Sphere: (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) is the center and r is the radius.
• Intersection with Plane: The sphere intersects the plane at a circle if the plane passes through the center of the sphere. Otherwise, it intersects at two points.
• Intersection with Line: The sphere intersects the line at two points if the line passes through the center of the sphere. Otherwise, it intersects at one point or no point at all.

Step-by-Step Problem-Solving Strategy

1. Identify the given information: Clearly understand the equation of the sphere, the equation of the plane/line, and any given points or lines.
2. Set up the equation: Use the equation of the sphere and the equation of the plane/line to find the points of intersection.
3. Check for special conditions: Verify if the plane passes through the center of the sphere or if the line passes through the center.
4. Solve for the intersection points: Use algebraic manipulations to find the coordinates of the intersection points.
5. Avoid common mistakes: ⚠️ Don't assume the plane intersects the sphere at a circle without verifying.

Important Graphs / Diagrams (if applicable)

• Sphere: A 3D representation of a sphere with its center and radius marked.
• Plane: A 2D representation of a plane with its equation and normal vector marked.
• Line: A 2D representation of a line with its equation and direction vector marked.

Typical JEE Question Patterns

• Find the equation of the sphere: Use the given points to find the equation of the sphere.
• Find the points of intersection: Use the equation of the sphere and the equation of the plane/line to find the points of intersection.
• Compare time periods: Compare the time periods of two spheres rolling down different inclined planes.

Common Mistakes & Exam Traps

• The mistake: ⚠️ Assuming the plane intersects the sphere at a circle without verifying.
• Why it happens: Misreading the equation of the plane or misunderstanding the properties of the sphere.
• How to avoid it: Verify if the plane passes through the center of the sphere before assuming intersection at a circle.

• Exam board insight: The examiners penalize this mistake by deducting marks for incorrect assumptions.

• The mistake: ⚠️ Not checking for special conditions.

• Why it happens: Rushing through the problem or not reading the question carefully.
• How to avoid it: Read the question carefully and verify the special conditions before solving.
• Exam board insight: The examiners penalize this mistake by deducting marks for incomplete solutions.

Time-Saving Shortcuts (if any)

• Use the equation of the sphere to find the distance from the center to the plane.
• Use the equation of the plane to find the distance from the center to the line.

Practice MCQs (Exam-Style)

Question 1: A sphere of radius r has its center at the origin. A plane passes through the point (2, 3, 4). Which of the following equations represents the plane? A) x + y + z = 0
B) x - y - z = 0
C) x + y - z = 0
D) x - y + z = 0

Answer: A) x + y + z = 0
Solution: The equation of the plane passing through the point (2, 3, 4) is x + y + z = 0.
Common Wrong Answer: Option B) x - y - z = 0 is tempting because it has a similar equation structure, but it does not pass through the point (2, 3, 4).

Question 2: A line passes through the point (1, 2, 3) and has a direction vector (2, 3, 4). A sphere of radius r has its center at the origin. Which of the following statements is true? A) The line intersects the sphere at two points.
B) The line intersects the sphere at one point.
C) The line does not intersect the sphere.
D) The line passes through the center of the sphere.

Answer: C) The line does not intersect the sphere.
Solution: The line does not intersect the sphere because the distance from the center to the line is greater than the radius of the sphere.
Common Wrong Answer: Option A) The line intersects the sphere at two points is tempting because it has a similar equation structure, but it does not pass through the center of the sphere.

Question 3: A sphere of radius r has its center at the point (a, b, c). A plane passes through the center of the sphere and has a normal vector (2, 3, 4). Which of the following equations represents the plane? A) 2x + 3y + 4z = 0
B) 2x - 3y - 4z = 0
C) x + y - z = 0
D) x - y + z = 0

Answer: A) 2x + 3y + 4z = 0
Solution: The equation of the plane passing through the center of the sphere is 2x + 3y + 4z = 0.
Common Wrong Answer: Option B) 2x - 3y - 4z = 0 is tempting because it has a similar equation structure, but it does not pass through the center of the sphere.

Quick Revision Card (60-Second Summary)

• Equation of a sphere: (x - a)² + (y - b)² + (z - c)² = r²
• Intersection with plane: Verify if the plane passes through the center of the sphere.
• Intersection with line: Verify if the line passes through the center of the sphere.
• Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
• Normal vector: (2, 3, 4) is a normal vector to the plane 2x + 3y + 4z = 0

If You Get Stuck in Exam

• Write down what you know: Even if unsure, write down the equation of the sphere, the equation of the plane/line, and any given points or lines.
• Eliminate distractors: Read the question carefully and eliminate any options that are clearly incorrect.
• Skip and return: If stuck, skip the question and return to it later with a fresh mind.

Related JEE Topics

• Coordinate Geometry: Understand the basics of coordinate systems, distance formula, and equations of lines and planes.
• Mathematical Reasoning: Familiarize yourself with mathematical induction, proof by contradiction, and logical deductions.
• Geometry: Know basic concepts of points, lines, planes, and solids, including properties of congruent and similar figures.