JEE Mathematics 3D Geometry Planes Equation Angle Between Planes Distance

3D Geometry — Planes: Equation, Angle Between Planes, Distance

What This Is and Why It Matters for JEE

Three-dimensional geometry is a crucial topic in JEE, appearing in 2-3 questions every year. It's moderately difficult and equally important for both Main and Advanced exams.

Prerequisites

You should already know: - Coordinate geometry (points, lines, planes) - Vectors (magnitude, direction, operations) - Trigonometry (angles, triangles, identities)

Quick revision path: Brush up on coordinate geometry and vectors.

Core Concepts (Exam-Focused)

Key concepts for JEE problems:

• Equation of a plane: ax + by + cz + d = 0, where a, b, c, and d are constants.
• Angle between planes: Use cos θ = |a1a2 + b1b2 + c1c2| / (√(a12 + b12 + c12) × √(a22 + b22 + c22)), where a1, b1, c1 and a2, b2, c2 are coefficients of the planes.
• Distance between a point and a plane: Use D = |ax1 + by1 + cz1 + d| / √(a2 + b2 + c2), where (x1, y1, z1) is the point and ax + by + cz + d = 0 is the plane.

Step-by-Step Problem-Solving Strategy

1. Identify the given information (point, plane, angle).
2. Check if the equation of the plane is given in normal form (ax + by + cz + d = 0).
3. If not, convert it to normal form.
4. Use the formula for the angle between planes or distance between a point and a plane.
5. ⚠️ Avoid using the formula for the angle between two planes if the planes are parallel or perpendicular.

Important Graphs / Diagrams (if applicable)

No specific graphs or diagrams are required for this topic.

Typical JEE Question Patterns

1. Find the equation of a plane passing through a given point and parallel to a given plane.
   • Recognition clue: "parallel" and "perpendicular".
   • Go-to method: Use the equation of the given plane and the point to find the equation of the parallel plane.
2. Find the angle between two given planes.
   • Recognition clue: "angle between planes".
   • Go-to method: Use the formula for the angle between planes.
3. Find the distance between a given point and a given plane.
   • Recognition clue: "distance between point and plane".
   • Go-to method: Use the formula for the distance between a point and a plane.

Common Mistakes & Exam Traps

1. The mistake: Using the formula for the angle between two planes when the planes are parallel or perpendicular.
   • Why it happens: Misreading the question or misunderstanding the concept.
   • How to avoid it: Check if the planes are parallel or perpendicular before using the formula.
   • Exam board insight: Examiners will penalise incorrect use of the formula.
2. The mistake: Not converting the equation of the plane to normal form.
   • Why it happens: Rushing through the problem or not checking the equation.
   • How to avoid it: Always check the equation of the plane and convert it to normal form if necessary.
3. The mistake: Not using the correct formula for the angle between planes or distance between a point and a plane.
   • Why it happens: Misreading the question or not checking the formula.
   • How to avoid it: Read the question carefully and check the formula before using it.

Time-Saving Shortcuts (if any)

None.

Practice MCQs (Exam-Style)

Question 1 (Easy)
Find the equation of the plane passing through the point (1, 2, 3) and parallel to the plane 2x - 3y + 4z - 5 = 0.

A) 2x - 3y + 4z + 2 = 0 B) 2x - 3y + 4z - 2 = 0 C) 2x - 3y + 4z + 5 = 0 D) 2x - 3y + 4z - 5 = 0

Answer: A) 2x - 3y + 4z + 2 = 0 Solution: Use the equation of the given plane and the point to find the equation of the parallel plane.
Common Wrong Answer: D) 2x - 3y + 4z - 5 = 0 (parallel plane with same equation).

Question 2 (Moderate)
Find the angle between the planes 2x - 3y + 4z - 5 = 0 and x + 2y - 3z + 4 = 0.

A) 30° B) 45° C) 60° D) 90°

Answer: B) 45° Solution: Use the formula for the angle between planes.
Common Wrong Answer: A) 30° (misreading the coefficients).

Question 3 (JEE Advanced)
Find the distance between the point (1, 2, 3) and the plane x + 2y - 3z + 4 = 0.

A) 5 B) 10 C) 15 D) 20

Answer: B) 10 Solution: Use the formula for the distance between a point and a plane.
Common Wrong Answer: A) 5 (misreading the point or plane).

Quick Revision Card (60-Second Summary)

• Equation of a plane: ax + by + cz + d = 0
• Angle between planes: cos θ = |a1a2 + b1b2 + c1c2| / (√(a12 + b12 + c12) × √(a22 + b22 + c22))
• Distance between a point and a plane: D = |ax1 + by1 + cz1 + d| / √(a2 + b2 + c2)
• Convert equation of plane to normal form if necessary.

If You Get Stuck in Exam

1. Write down what you know and what you need to find.
2. Eliminate distractors by checking the options.
3. If stuck, skip and return to the question later.

Related JEE Topics

• Coordinate geometry: points, lines, circles
• Vectors: magnitude, direction, operations
• Trigonometry: angles, triangles, identities