NCERT Class 11 Mathematics Mathematics: Chapter 10 — MATHEMATICS
This chapter, "Conic Sections," introduces students to curves formed by the intersection of a plane and a double-napped right circular cone. It builds upon the equations of lines studied previously. The chapter defines key terms like vertex, axis, and generator, and explains how different conic sections – circles, ellipses, parabolas, and hyperbolas – are formed based on the angle of the intersecting plane relative to the cone's axis. These curves have significant applications in various scientific and technological fields, including astronomy and optics. The chapter aims to provide a foundational understanding of these geometric shapes and their formation, preparing students for further study in mathematics and its applications.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 11 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Chapter 10 — MATHEMATICS |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 3 minutes |
| Word count | 515 |
Learning outcomes
- Define conic sections and identify their relationship with a double-napped right circular cone.
- Understand the terms vertex, axis, and generator of a cone.
- Differentiate between circle, ellipse, parabola, and hyperbola based on the intersecting plane's angle.
- Recognize the wide applications of conic sections in real-world scenarios.
Vocabulary
| Word | Meaning |
|---|---|
| Conic Sections | Curves obtained by the intersection of a plane and a double-napped right circular cone. |
| Vertex | The fixed point where the axis intersects the cone. |
| Axis | The fixed vertical line around which the generator rotates to form the cone. |
| Generator | The rotating line that forms the surface of the cone. |
| Nappes | The two parts of the cone separated by the vertex. |
| Ellipse | A conic section formed when the intersecting plane's angle (β) is between the generator's angle (α) and 90 degrees. |
| Parabola | A conic section formed when the intersecting plane's angle (β) is equal to the generator's angle (α). |
| Hyperbola | A conic section formed when the intersecting plane's angle (β) is less than the generator's angle (α) and cuts through both nappes. |
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Practice questions
- What is a conic section? Answer: A conic section is a curve obtained by the intersection of a plane with a double-napped right circular cone.
- What are the two parts of a cone called? Answer: The two parts of a cone are called nappes.
- Under what condition is the conic section a circle? Answer: The conic section is a circle when the intersecting plane is perpendicular to the axis (β = 90°).
- When does the intersection of a plane and a cone result in a parabola? Answer: The intersection results in a parabola when the angle made by the intersecting plane with the vertical axis (β) is equal to the angle made by the generator with the vertical axis (α).
Practice MCQs
Q1. Which of the following is NOT a conic section?
Explanation: Circle, ellipse, parabola, and hyperbola are the primary conic sections formed by the intersection of a plane and a cone. A square is not typically formed this way.
Q2. What is the term for the surface generated by rotating a line around a fixed axis?
Explanation: The rotation of a line (generator) around a fixed line (axis) generates a cone.
Q3. If the angle β made by the intersecting plane with the vertical axis is 90°, what conic section is formed?
Explanation: When β = 90°, the intersecting plane is perpendicular to the axis, resulting in a circle.
Q4. A hyperbola is formed when the intersecting plane cuts through:
Explanation: A hyperbola is formed when the intersecting plane cuts through both nappes of the cone, with the angle β being less than α.
Q5. Who is credited with naming the parabola and hyperbola?
Explanation: The text mentions that Apollonius gave the names parabola and hyperbola.
Frequently asked questions
What are conic sections?
Conic sections are curves formed by the intersection of a plane with a double-napped right circular cone. The common types are circles, ellipses, parabolas, and hyperbolas.
What is a double-napped right circular cone?
It's a cone formed by rotating a line (generator) around a fixed vertical line (axis), extending indefinitely in both directions, creating two identical halves called nappes.
How is a circle formed as a conic section?
A circle is formed when the intersecting plane is perpendicular to the cone's axis (angle β = 90°).
What is the condition for forming an ellipse?
An ellipse is formed when the angle of the intersecting plane (β) is greater than the angle of the generator (α) but less than 90° (α < β < 90°).
When does the intersection result in a parabola?
A parabola is formed when the angle of the intersecting plane (β) is exactly equal to the angle of the generator (α) (β = α).
What is required to form a hyperbola?
A hyperbola is formed when the angle of the intersecting plane (β) is less than the angle of the generator (α) (0 ≤ β < α), and the plane cuts through both nappes of the cone.
What are some real-life applications of conic sections?
Conic sections have applications in planetary motion, designing telescopes and antennas, and in reflectors for flashlights and headlights.
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NCERT Class 11 Mathematics — Mathematics — Chapter 10 — MATHEMATICS. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.