Lesson No. 1 Topic: The Conics I. Objectives: At the end of the lessons, the learners shall be able to: 1. Illustrate the different types of conic sections – circle, parabola, ellipse, and hyperbola. 2. Determine the type of conic section defined by a given 2nd degree equation in x and y. 3. Appreciate the importance of the conic sections to different real life situations. II. Key Concepts The Conics A conic section is the intersection of a plane and a right circular cone or two right circular cones with a common vertex and parallel bases. Consider the two – napped right circular cone in Figure 1. A line on the cone is called a generator and the only point where the generators intersect is called the vertex. The vertical line passing through the vertex is called the axis of the cone. The angular orientation of the plane relative to the cone defines whether the conic section is a circle, parabola, ellipse or hyperbola. The circle and the ellipse are formed when the intersection of cone and plane is abounded curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. The ellipse is formed when the tilted plane intersects only one cone to form a bounded curve. If the plane is parallel to a generator line of the cone, the conic is a parabola. Lastly, if the intersection is an unbounded curve and the plane is not parallel to a generator line of the cone and the plane intersects both halves of the cone, the conic is a hyperbola. These are illustrated in Figure 2. The graph of the second – degree equation of the form 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 is determined by the values of 𝐵2 − 4𝐴𝐶. Table 1 below provides the type of conic section given the values of A, B and C. Conic sections can also be defined in terms of eccentricity e. Table 1: Quadratic Relations and Conic Section Conic Section Circle Parabola Ellipse Hyperbola Value of 𝐵2 − 4𝐴𝐶 𝐵2 − 4𝐴𝐶 < 0, 𝐵 = 0 𝑜𝑟 𝐴 = 𝐶 𝐵2 − 4𝐴𝐶 = 0 𝐵2 − 4𝐴𝐶 < 0, 𝐵 ≠ 0 𝑜𝑟 𝐴 ≠ 𝐶 𝐵2 − 4𝐴𝐶 > 0 Eccentricity 𝑒=0 𝑒=1 0<𝑒<1 𝑒>1 Example : Determine the type of conic section that each general equation will produce. Procedure: Collect first all the values of A, B and C and solve for the value of 𝐵2 − 4𝐴𝐶. Then interpret the result based on Table 1. a. 𝟗𝒙𝟐 + 𝟗𝒚𝟐 − 𝟔𝒙 + 𝟏𝟖𝒚 + 𝟏𝟏 = 𝟎 𝐴 = 9, 𝐵 = 0, 𝐶 = 9 𝐵2 − 4𝐴𝐶 = 02 − 4(9)(9) = 0 − 324 = −324 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 − 324 < 0 Note also that B = 0 and A = C. Thus, the conic section is a circle. b. 𝟐𝒙𝟐 + 𝟒𝒚𝟐 + 𝟖𝒙 + 𝟐𝟒𝒚 + 𝟒𝟒 = 𝟎 𝐴 = 2, 𝐵 = 0, 𝐶 = 4 𝐵2 − 4𝐴𝐶 = 02 − 4(2)(4) = 0 − 32 = −32 𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 − 32 < 0 Note also that B ≠ 0 and A ≠ C. Thus, the conic section is an ellipse. c. 𝟒𝒙𝟐 + 𝟒𝒙𝒚 + 𝒚𝟐 + 𝟖𝒙 + 𝟐𝟒𝒚 + 𝟑𝟔 = 𝟎 𝐴 = 4, 𝐵 = 4, 𝐶 = 1 𝐵2 − 4𝐴𝐶 = 42 − 4(4)(1) = 16 − 16 =0 Thus, the conic section is a parabola. d. 𝟒𝒙𝟐 + 𝟔𝒙𝒚 + 𝟐𝒚𝟐 − 𝟒𝒙 − 𝟐𝒚 + 𝟒𝟑 = 𝟎 𝐴 = 4, 𝐵 = 6, 𝐶 = 2 𝐵2 − 4𝐴𝐶 = 62 − 4(4)(2) = 36 − 32 =4 Note that 4 0 Thus, the conic section is a hyperbola. Conics are very important in our daily life for some of the following reasons: 1. The world is round. 2. There are situations that are cyclic in nature. 3. The paths of the planets around the sun are ellipses with the sun at one focus. 4. The parabola is used in the design of car headlights and in spotlights because it aids in concentrating the light beam. 5. The ideas of ellipse and hyperbola serve as the student’s foundation in the elliptical and hyperbolic turns and motion. Several applications in the fields of astronomy and design use the ellipse and the hyperbola as the serving model. For further readings: https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)/11%3A_Parametric_Eq uations_and_Polar_Coordinates/11.5%3A_Conic_Sections https://math.uh.edu/~shanyuji/History/h-11.pdf https://www.purplemath.com/modules/conics.htm https://www.slideshare.net/farhanashaheen1/conic-section-ppt