Conic Sections | Class 11 Mathematics Notes

10.1 Introduction

In this chapter, we begin our study of conic sections, a fascinating family of curves that arise from the intersection of a plane with a double-napped right circular cone. These curves include the circle, ellipse, parabola, and hyperbola. The term 'conic sections' or 'conics' is derived from this geometric origin. The study of these curves is not only theoretically important in mathematics but also has wide-ranging applications in physics, astronomy, engineering, and other fields. For example, planetary orbits are ellipses, parabolic reflectors are used in telescopes and antennas, and hyperbolas appear in navigation and signal processing. The names parabola and hyperbola were given by the ancient Greek mathematician Apollonius, who made significant contributions to the understanding of these curves. This chapter will explore the definitions, equations, properties, and applications of these conic sections, starting from their geometric definitions and moving towards their algebraic representations.

📊 Diagram: Figure 1 on page 1; Figure 2 on page 1; Figure 3 on page 1; Figure 4 on page 1; Table on page 1 (2×2)

🧪 Activity: No specific activity in this introductory section.

🔗 Connection: This introduction sets the stage for the detailed study of each conic section starting with the circle in the next section.

Table on page 1 (2×2)

| v Let the relation of knowledge to real life be very visible to your pupils and let them understand how by knowledge the world could be v transformed. – BERTRAND RUSSELL 10.1 Introduction In the preceding Chapter 10, we have studied various forms of the equations of a line. In this Chapter, we shall study about some other curves, viz., circles, ellipses, parabolas and hyperbolas. The names parabola and hyperbola are given by Apollonius. These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with a double napped right circular cone. These curves have a very wide range of applications in fields such as planetary motion, Apollonius design of telescopes and antennas, reflectors in flashlights (262 B.C. -190 B.C.) and automobile headlights, etc. Now, in the subsequent sections we will see how the intersection of a plane with a double napped right circular cone results in different types of curves. 10.2 Sections of a Cone Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α (Fig10.1). | | | Suppose we rotate the line m around the line l in such a way that the angle α remains constant. Then the surface generated is | |

Table on page 20 (3×5)

| In each of the Exercises 1 to 9, find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse. x2 y2 x2 y2 x2 y2 1. + =1 2. + =1 3. + =1 36 16 4 25 16 9 x2 y2 x2 y2 x2 y2 4. + =1 5. + =1 6. + = 1 25 100 49 36 100 400 7. 36x2 + 4y2 = 144 8. 16x2 + y2 = 16 9. 4x2 + 9y2 = 36 In each of the following Exercises 10 to 20, find the equation for the ellipse that satisfies the given conditions: 10. Vertices (± 5, 0), foci (± 4, 0) 11. Vertices (0, ± 13), foci (0, ± 5) 12. Vertices (± 6, 0), foci (± 4, 0) 13. Ends of major axis (± 3, 0), ends of minor axis (0, ± 2) 14. Ends of major axis (0, ± 5 ), ends of minor axis (± 1, 0) 15. Length of major axis 26, foci (± 5, 0) 16. Length of minor axis 16, foci (0, ± 6). 17. Foci (± 3, 0), a = 4 18. b = 3, c = 4, centre at the origin; foci on the x axis. 19. Centre at (0,0), major axis on the y-axis and passes through the points (3, 2) and (1,6). 20. Major axis on the x-axis and passes through the points (4,3) and (6,2). | | | | |