EQUATION AND IMPORTANT CHARACTERISTICS OF THE DIFFERENT TYPES OF CONIC SECTIONS for Pre Calculus Senior High School (STEM) Quarter 1 / Week 5 1 FOREWORD This Self-Learning Kit (SLK) will serve as a guide in studying the subject Area Pre-Calculus. Guided by the Most Essential Learning Competencies (MELC), it will be used as an aid in learning the concepts of conic sections and systems of nonlinear equation. Moreover, it gives focus on the knowledge, understanding, skills , and attitudes that need to be demonstrated in this lesson. What happened (Let’s Do It) This section contains pre-activities like review of the prior knowledge of the different types of conic sections. What I Need to Know (Discussion) This section contains discussion on how to recognize the equation and important characteristics of the different types of conic sections wherein students will apply their mathematical concepts learned. What I have Learned (Evaluation/Post-Test) The exercises contained in this section enhanced student’s comprehension and mathematical skills. These serve as a diagnostic tool to identify student’s weaknesses and strengths. 2 OBJECTIVES: K. Identify the important characteristics of the equation of the different types of conic sections. S. Classify equations of the conic sections into circle, parabola, hyperbola, or ellipse. A. Appreciate the importance of the characteristics of the different types of conic section to real life situation. LEARNING COMPETENCY: Recognize the equation and important characteristics of the different types of conic sections (STEM_PC11AG – Ie – 1). I. What Happened PRE-TEST: A. Classify the following equations whether circle, parabola, ellipse or hyperbola. Answer in your activity sheet/notebook. ______ 1. 3x2 – 12x + 3y2 = 2 ______ 2. y = x2 – 4 ______ 3. 3x2 – 9x + 2y2 +10y – 6 =0 ______ 4. 4y2 – 10y - 3x2 = 12 ______ 5. X = 2y2 – 3y + 10 3 II. What You Need to Know In this lesson, we will discuss on how to recognize the equatio In this lesson, we will discuss on how to recognize the equation and also know the important characteristics of the different types of conic section. DISCUSSION: A. Recognizing the Equation The equation of a circle is written in a standard form Ax2 + Ay2 – Cx + Dy +E = 0, That is, the coefficients of x2 and y2 are the same. However, it does not follow that if the coefficients of x2 and y2 are the same, the graph is a circle. General Equation Standard Equation 1 3 Graph A. 2x2 + 2y2 – 2x + 6y + 5 = 0 (x- 2)2 + (y +2)2 = 0 point B. x2 + y2 – 6x + 8y + 50 = 0 (x - 3)2 + (y - 4)2 = -25 empty set For a circle with equation (x – h)2 + (y – k)2 = r2, we have r2 > 0. This is not the case for the standard equations of (A) and (B). 4 In (A), because the sum of two squares can only be 0 if and only if each 1 3 square is 0, it follows that x− 2= 0 and y + 2 = 0. The graph is thus the single point 1 3 (2, −2). In (B), no real values of x and y can make the nonnegative left side equal to the negative right side. The graph is then the empty set. Example: General Form to Standard form. 2x 2 + 2 y 2 − 2x + 6 y + 5 = 0 General Form (2 x − 2 x) (2 y + 6 y ) − 5 + = 2 2 2 −5 ( x 2 − x) + ( y 2 + 3 y ) = 2 1 9 −5 1 9 (x 2 − x + ) + ( y 2 + 3y + ) = + + 4 4 2 4 4 1 9 − 10 + 1 + 9 (x 2 − x + ) + ( y 2 + 3y + ) = 4 4 4 1 9 0 (x 2 − x + ) + ( y 2 + 3y + ) = 4 4 4 1 3 (x − )2 + ( y + )2 = 0 Standard Form 2 2 2 2 B. Important characteristics of conic section Let us recall the general form of the equations of the other conic sections. We may write the equations of conic sections we discussed in the general form Ax2 + By2 + Cx + Dy + E = 0. Some terms may vanish, depending on the kind of conic section. (1) Circle: both x2 and y2 appear, and their coe๏ฌcients are the same Ax2 + Ay2 + Cx + Dy + E = 0 5 Example: 18x2 + 18y2 −24x + 48y−5 = 0 Degenerate cases: a point, and the empty set (2) Parabola: exactly one of x2 or y2 appears Ax2 + Cx + Dy + E = 0 (D 6= 0, opens upward or downward) By2 + Cx + Dy + E = 0 (C 6= 0, opens to the right or left) Examples: 3x2 −12x + 2y + 26 = 0 (opens downward) −2y2 + 3x + 12y−15 = 0 (opens to the right) (3) Ellipse: both x2 and y2 appear, and their coe๏ฌcients A and B have the same sign and are unequal Examples: 2x2 + 5y2 + 8x−10y−7 = 0 (horizontal major axis) 4x2 + y2 −16x−6y + 21 = 0 (vertical major axis) If A = B, we will classify the conic as a circle, instead of an ellipse. Degenerate cases: a point, and the empty set (4) Hyperbola: both x2 and y2 appear, and their coe๏ฌcients A and B have different signs Examples: 5x2 −3y2 −20x−18y−22 = 0 (horizontal transverse axis) −4x2 + y2 + 24x + 4y−36 = 0 (vertical transverse axis) Degenerate case: two intersecting lines (4) Hyperbola: both x2 and y2 appear, and their coe๏ฌcients A and B have different signs Examples: 5x2 −3y2 −20x−18y−22 = 0 (horizontal transverse axis) −4x2 + y2 + 24x + 4y−36 = 0 (vertical transverse axis) Degenerate case: two intersecting lines The following examples will show the possible degenerate conic (a point, two intersecting lines, or the empty set) as the graph of an equation following a similar pattern as the non-degenerate cases. (๐ฅ−2)2 (1) 4x2 + 9y2 −16x + 18y + 25 = 0 (๐ฆ + 1)2 =⇒ 32 + 22 = 0 =⇒ one point: (2, −1) 6 (๐ฅ−2)2 (๐ฆ + 1)2 (2) 4x2 + 9y2 −16x + 18y + 61 = 0 =⇒ 32 + 22 =⇒ empty set (3) 4x2 −9y2 −16x−18y + 7 = 0 =⇒ (๐ฅ−2)2 32 - (๐ฆ + 1)2 22 = -1 =0 2 =⇒ two lines: y + 1 = ± 3(x−2) A Note on Identifying a Conic Section by Its General Equation It is only after transforming a given general equation to standard form that we can identify its graph either as one of the degenerate conic sections (a point, two intersecting lines, or the empty set) or as one of the non-degenerate conic sections (circle, parabola, ellipse, or hyperbola). 1. Parabola The parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. The equation for a parabola is y=a(x−b)2+c or x =a(y−b)2+c 2. Circles and Ellipses The equation of a circle with center at (a, b) and radius r units is (x−a) 2 + (y−b) 2 = r2 An ellipse is the figure consisting of all points in the plane whose coordinates satisfy the equation ๐ฑ๐ ๐ฒ๐ + =1 ๐๐ ๐ ๐ If the ellipse has its center at (m, n) the equation could be written as (๐ฑ−๐ฆ)๐ ๐๐ + (๐ฒ−๐ง)๐ ๐๐ 7 =1 3. Hyperbolas A hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror images of each other and resembling two infinite bows. The equation of a hyperbola with a center at (m, n) is (x−m)2/a2 − (y−n)2/b2 = 1 (๐ฑ−๐ฆ)๐ ๐๐ - (๐ฒ−๐ง)๐ ๐๐ =1 III. What Have I Learned POST TEST: The graphs of the following equations are (nondegenerate) conic sections. Identify the conic section. Write your answer in your activity sheet/notebook. (1) 5x2 −3y2 + 10x−12y = 22 (2) 2y2 −5x−12y = 17 (3) 3x2 + 3y2 + 42x−12y = −154 (4) 3x2 + 6x + 4y = 18 (5) 7x2 + 3y2 −14x + 12y = −14 (6) −4x2 + 3y2 + 24x−12y = 36 8 DEPARTMENT OF EDUCATION SCHOOLS DIVISION OF NEGROS ORIENTAL SENEN PRISCILLO P. PAULIN, CESO V Schools Division Superintendent FAY C. LUAREZ, TM, Ed.D., Ph.D. OIC - Assistant Schools Division Superintendent Acting CID Chief ADOLF P. AGUILAR OIC - Assistant Schools Division Superintendent NILITA L. RAGAY, Ed.D. OIC - Assistant Schools Division Superintendent ROSELA R. ABIERA Education Program Supervisor – (LRMS) ARNOLD R. JUNGCO Education Program Supervisor – (SCIENCE & MATH) MARICEL S. RASID Librarian II (LRMDS) ELMAR L. CABRERA PDO II (LRMDS) REONELL K. SELARDE Writer LITTIE BETH S. BERNADEZ Lay-out Artist _________________________________ ALPHA QA TEAM LITTIE BETH S. BERNADEZ MERCYDITHA D. ENOLPE RONALD TOLENTINO DIDITH T. YAP BETA QA TEAM ELIZABETH A. ALAP-AP EPIFANIA Q. CUEVAS NIDA BARBARA S. SUASIN VRENDIE P. SYGACO MELBA S. TUMARONG HANNAHLY I. UMALI DISCLAIMER The information, activities and assessments used in this material are designed to provide accessible learning modality to the teachers and learners of the Division of Negros Oriental. The contents of this module are carefully researched, chosen, and evaluated to comply with the set learning competencies. The writers and evaluator were clearly instructed to give credits to information and illustrations used to substantiate this material. All content is subject to copyright and may not be reproduced in any form without expressed written consent from the division. 9 REFERENCES Bacani, Jerico B., Eden, Richard B., Estrada, Glenn Rey A., Francisco, Flordeliza F., Vidallo, Mark Anthony J., Department of Education, Ground Floor Bonifacio Building, Dep.Ed Complex Meralco Avenue, Pasig City, Philippines 1600, 2016, Page 60-62. https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/conic-sections/the-fourconic-sections https://www.mathplanet.com/education/algebra-2/conic-sections/equations-ofconic-sections https://www.dummies.com/education/math/calculus/how-to-identify-the-four-conicsections-in-equation-form/ 10 SYNOPSIS AND ABOUT THE AUTHOR SYNOPSIS The Self Learning Kit (SLK) is developed to prepare students explore mathematical problems involving conic sections. Students are provided with practice activities, examples and assessment to test their skills and knowledge, sharpen their critical thinking skills and lead them to develop wise decision making. ANSWERS KEY Pre activities / Pre-test A. 1. Circle 2. Parabola 3. Ellipse 4. Hyperbola 5. Parabola Evaluation/ Post test 1. Hyperbola 2. Parabola 3. Circle 4. Parabola 5. Ellipse 6. Hyperbola AUTHOR Reonell K. Selarde is a graduate of Bachelor of Science in Industrial Engineering at Foundation University, Dumaguete City. Also finished his Continuing Professional Education at Presbyterian Theological College Inc., Dumaguete City. He is a Grade 12 teacher-adviser Maria Macahig Memorial High School, Siaton, Negros oriental and has been serving the Department of Education since 2018. 11