Conic Sections

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. **Table on page 1 (2×2)** | CONIC SECTIONS | | | --- | --- | | 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)** | | | | EXERCISE 10.3 | | | --- | --- | --- | --- | --- | | 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). | | | | | | | | | | | | | 10.6 | Hyperbola | | |

• Conic sections are curves obtained by intersecting a plane with a double-napped right circular cone.
• The four main types of conics are circle, ellipse, parabola, and hyperbola.
• These curves have important applications in planetary motion, telescope design, antennas, and reflectors.
• The names parabola and hyperbola were introduced by Apollonius.
• The chapter will cover geometric definitions and algebraic equations of conics.
• 📌 Conic sections: Curves formed by the intersection of a plane with a double-napped right circular cone.
• 📌 Double-napped cone: A cone with two identical parts (nappes) extending infinitely in opposite directions.
• 📌 Apollonius: Ancient Greek mathematician who studied conic sections.

10.2 Circle

A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the centre. The fixed distance from the centre to any point on the circle is called the radius. The standard equation of a circle with centre at (h, k) and radius r is (x – h)² + (y – k)² = r². This equation is derived using the distance formula between any point (x, y) on the circle and the centre (h, k). When the centre is at the origin (0, 0), the equation simplifies to x² + y² = r². The chapter also discusses how to find the equation of a circle given its centre and radius, or given three points on the circle. Examples include finding the equation of a circle with centre (–3, 2) and radius 4, and completing the square to find the centre and radius from a general quadratic equation. The circle is a fundamental conic with constant distance property, and its properties form the basis for understanding more complex conics.

• Circle is the locus of points equidistant from a fixed point called the centre.
• Radius is the fixed distance from the centre to any point on the circle.
• Standard equation of a circle with centre (h, k) and radius r is (x – h)² + (y – k)² = r².
• If the centre is at the origin, the equation reduces to x² + y² = r².
• The distance formula is used to derive the equation of a circle.
• Completing the square helps to find the centre and radius from a general quadratic equation.
• 📌 Centre of circle: Fixed point from which all points on the circle are equidistant.
• 📌 Radius: The constant distance from the centre to any point on the circle.