Lesson 1: Conic Sections PDF

OUR LADY OF FATIMA UNIVERSITY SY: 2021 – 2022 (SECOND SEMESTER)  COEFFICIENT CONSTANT 𝟐 𝟐 𝟐𝒙 − 𝟑𝒚 + 𝟐𝒙 − 𝒚 + 𝟐𝟐 = 𝟎 TERM VARIABLE 2ND DEGREE EQUATION Definition: A conic is defined as a set of curves formed from dividing or cutting a right circular cone. These four typ...

OUR LADY OF FATIMA UNIVERSITY SY: 2021 – 2022 (SECOND SEMESTER)  COEFFICIENT CONSTANT 𝟐 𝟐 𝟐𝒙 − 𝟑𝒚 + 𝟐𝒙 − 𝒚 + 𝟐𝟐 = 𝟎 TERM VARIABLE 2ND DEGREE EQUATION Definition: A conic is defined as a set of curves formed from dividing or cutting a right circular cone. These four types of curves are: circle, ellipse, parabola and hyperbola. The GENERAL FORM of a conic is: 𝟐 𝟐 𝑨𝒙 + 𝑩𝒙𝒚 + 𝑪𝒚 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 Definition: If the plane cuts only one Definition: If the plane cuts only one of the circular cone nappes and it is of the circular cone nappes and is perpendicular to the axis, the formed parallel to the surface base of the image is circle. cone with an angle, the formed image is ellipse. Definition: If the cutting plane is Definition: If the plane cuts both parallel only to one generator and it nappes and is parallel to two is perpendicular to the base of the generators, the intersection curved is cone, the curved is formed called a hyperbola. parabola.  𝟐 𝑩 − 𝟒𝑨𝑪  Quadratic Terms and Type 𝟐 𝑩 − 𝟒𝑨𝑪 Constants 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH CIRCLE 𝐵 2 − 4𝐴𝐶 < 0 where 𝐵 = 0 𝑜𝑟 𝐴 = 𝐶 present 𝐴=𝐶 Hb: What type of a curve is the equation: 𝒙 𝟐 + 𝒚𝟐 − 𝟑𝒙 + 𝟒 = 𝟎 A. By the formula: 𝐵2 − 4𝐴𝐶 B. By observing the quadratic terms Terms 𝐴𝑥 2 = 𝑥 2 and 𝐶𝑦 2 = 𝑦 2 are both present. Also, 𝐴 = 1 and 𝐶 = 1, therefore 𝐴 = 𝐶. By its given quadratic terms, this equation is a circle.  Quadratic Terms and Type 𝟐 𝑩 − 𝟒𝑨𝑪 Constants 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH ELLIPSE 𝐵 2 − 4𝐴𝐶 < 0 where 𝐵 = 0 𝑜𝑟 𝐴 ≠ 𝐶 present 𝐴≠𝐶 Hb: What type of a curve is the equation: 𝟑𝒙 𝟐 − 𝟗𝒙 = −𝟐𝒚𝟐 − 𝟏𝟎𝒚 + 𝟔 A. By the formula: 𝐵2 − 4𝐴𝐶 B. By observing the quadratic terms Terms 𝐴𝑥 = 3𝑥 and 𝐶𝑦 = 2𝑦 2 2 2 2 are both present. Also, 𝐴 = 3 and 𝐶 = 2, therefore 𝐴 ≠ 𝐶. By its given quadratic terms, this equation is a ellipse.  Quadratic Terms and Type 𝟐 𝑩 − 𝟒𝑨𝑪 Constants Only ONE quadratic term is PARABOLA 𝐵2 − 4𝐴𝐶 = 0 present (either 𝐴𝑥 2 or 𝐶𝑦 2 ) Hb: What type of a curve is the equation: 𝟐 𝒙 − 𝟑𝒙 − 𝒚 + 𝟕 = 𝟎 A. By the formula: 𝐵2 − 4𝐴𝐶 B. By observing the quadratic terms Term 𝐴𝑥 = 𝑥 is the only one 2 2 present. By its given quadratic terms, this equation is a parabola.  Quadratic Terms and Type 𝟐 𝑩 − 𝟒𝑨𝑪 Constants 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH HYPERBOLA 𝐵2 − 4𝐴𝐶 > 0 where 𝐴≠𝐶 present 𝐴≠𝐶 Hb: What type of a curve is the equation: 𝟐𝒙 𝟐 − 𝟑𝒚𝟐 + 𝟐𝒙 − 𝒚 + 𝟐𝟐 = 𝟎 A. By the formula: 𝐵2 − 4𝐴𝐶 B. By observing the quadratic terms Terms 𝐴𝑥 2 = 2𝑥 2 and 𝐶𝑦 2 = −3𝑦 2 are both present. Also, 𝐴 = 2 and 𝐶 = −3, therefore 𝐴 ≠ 𝐶. By its given quadratic terms, this equation is a hyperbola.  Quadratic Terms and Type 𝑩𝟐 − 𝟒𝑨𝑪 Constants 𝐵2 − 4𝐴𝐶 < 0 where 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH present CIRCLE 𝐵 = 0 𝑜𝑟 𝐴 = 𝐶 𝐴=𝐶 𝐵2 − 4𝐴𝐶 < 0 where 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH present ELLIPSE 𝐵 = 0 𝑜𝑟 𝐴 ≠ 𝐶 𝐴≠𝐶 Only ONE quadratic term is PARABOLA 𝐵2 − 4𝐴𝐶 = 0 present (either 𝐴𝑥 2 or 𝐶𝑦 2 ) 𝐵2 − 4𝐴𝐶 > 0 where 𝐴𝑥 2 and 𝐶𝑦 2 are BOTH present HYPERBOLA 𝐴≠𝐶 𝐴≠𝐶 Always transform the equation to GENERAL EQUATION first.