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Chapter 10 Conics, Parametric Equations, and Polar Coordinates Lecture Note Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler’s Laws Lecture Note Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler’s Laws Lecture Note 10.1 Conics and Calculus Geometric Definitions of Conic Sections and Their Standard Equations Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. There are seven different possible intersections. Three Others When the intersecting plane passes through the vertex of the cone. Four Basic Conics For four basic conics, the intersecting plane does not pass through the vertex of the cone. Lecture Note 10.1 Conics and Calculus Geometric Definitions of Conic Sections and Their Standard Equations Parabolas, ellipses, and hyperbolas 1. The paths of the planets around the sun (ellipses) 2. Parabolic mirrors are used to converge light beams at the focus of the parabola 3. Parabolic microphones perform a similar function with sound waves 4. Parabola is used in the design of car headlights and in spotlights 5. Trajectory of objects thrown or shot near the earth’s surface will follow a parabolic path 6. Hyperbolas are used in a navigation system known as LORAN(long range navigation) General second-degree equation 7. Hyperbolic as well as parabolic 2 2 퐴푥 + 퐵푥푦 + 퐶푦 + 퐷푥 + 퐸푦 + 퐹 = 0 mirrors and lenses are used in systems of telescopes. Lecture Note 10.1 Conics and Calculus One of approaches to define the conics is that the conics is defined as a locus (collection) of points satisfying a certain geometric property. For example, a circle can be defined as the collection of all points (푥, 푦) that are equidistant from a fixed point (ℎ, 푘). This locus definition easily produces the standard equation of a circle 푥 − ℎ 2 + 푦 − 푘 2 = 푟2. Lecture Note 10.1 Conics and Calculus Definition of Parabolas A parabola is a set of points in a plane that are equidistance from a fixed point 퐹 (called the focus) and a fixed line (called the directrix). The point halfway between the focus and the directrix lies on the parabola and it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. (0, 푝) Standard Equation 퐹 (focus) (푥, 푦) An equation of the parabola with focus (0, 푝) and directrix 푦 = −푝 is (0,0) 푥2 = 4푝푦 (See Figure) directrix 푦 = −푝 vertex axis Lecture Note 10.1 Conics and Calculus (0, 푝) (푥, 푦) Standard Equation 퐹 (focus) An equation of the parabola with focus (0, 푝) and directrix 푦 = −푝 is (0,0) 푥2 = 4푝푦 (See Figure) directrix 푦 = −푝 vertex axis vertex Standard Equation (푝, 0) An equation of the parabola with focus (푝, 0) and directrix (0,0) 퐹 (focus) axis 푥 = −푝 is 푦2 = 4푝푥 (푥, 푦) directrix 푥 = −푝 Lecture Note 10.1 Conics and Calculus Theorem 10.1 (Standard Equation of a Parabola) The standard form of the equation of a parabola with vertex (ℎ, 푘) and directrix 푦 = 푘 − 푝 is 푥 − ℎ 2 = 4푝 푦 − 푘 푉푒푟푡푐푎푙 푎푥푠 For directrix 푥 = ℎ − 푝, the equation is 푦 − 푘 2 = 4푝 푥 − ℎ 퐻표푟푧표푛푡푎푙 푎푥푠 The focus lies on the axis 푝 units (directed distance) from the vertex. The coordinates of the focus are as follows ℎ, 푘 + 푝 푉푒푟푡푐푎푙 푎푥푠 ℎ + 푝, 푘 퐻표푟푧표푛푡푎푙 푎푥푠 Example 1. Find the focus of the parabola 1 1 푦 = − 푥 − 푥2 2 2 Lecture Note 10.1 Conics and Calculus The Latus Rectum and Graphing Parabolas The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. The length of the latus rectum for the graphs of 푦2 = 4푝푥 and 푥2 = 4푝푦 is |4푝|. 푝 푝 푝 푝 2푝 4푝 Example 2. Find the length of the latus rectum of the parabola 푥2 = 4푝푦. Then find the length of the parabolic arc intercepted by the latus rectum. Lecture Note 10.1 Conics and Calculus Theorem 10.2 (Reflective Property of a Parabola) Let 푃 be a point on a parabola. The tangent line to the parabola at point 푃 makes equal angles with the following two lines 1. The line passing through 푃 and the focus. 2. The line passing through 푃 parallel to the axis of the parabola. Lecture Note 10.1 Conics and Calculus Ellipse An ellipse is the set of points in a plane the sum of whose distances from two fixed points 퐹1 and 퐹2 is a constant. These two fixed points are called the foci (plural of focus). The midpoint of the line segment connecting the foci is called the center. The line passing through two foci is called the major axis. The perpendicular line to the major axis passing through the midpoint of two foci is called the minor axis. The two intersection points of the major axis and the ellipse are called vertices. minor axis (0, 푏) 푃(푥, 푦) Standard Equation vertex vertex Let the sum of the distances from a point on the ellipse to the foci be 2푎. Then (−푎, 0) (0,0) (푎, 0) 퐹1 퐹2 푥2 푦2 (−푐, 0) (푐, 0) major axis + = 1 푎2 푏2 푐2 = 푎2 − 푏2 < 푎2. (0, −푏) where Lecture Note 10.1 Conics and Calculus minor axis (0, 푏) Standard Equation 푃(푥, 푦) Let the sum of the distances from vertex vertex a point on the ellipse to the foci be 2푎. Then 2 2 (−푎, 0) (0,0) (푎, 0) 푥 푦 퐹1 퐹2 + = 1 (−푐, 0) (푐, 0) major axis 푎2 푏2 where 푏2 = 푎2 − 푐2 < 푎2. (0, −푏) major axis (0, 푎) vertex Standard Equation (0, 푐) Let the sum of the distances from 퐹 1 푃(푥, 푦) a point on the ellipse to the foci (−푏, 0) be 2푎. Then (푏, 0) 푥2 푦2 (0,0) + = 1 minor axis 푏2 푎2 퐹 푏2 = 푎2 − 푐2 < 푎2. (0, −2푐) where vertex (0, −푎) Lecture Note 10.1 Conics and Calculus Theorem 10.3 Standard Equation of an Ellipse with Translations The standard form of the equation of a parabola with center (ℎ, 푘) and major and minor axes of lengths 2푎 and 2푏, 푎 > 푏, is 푥 − ℎ 2 푦 − 푘 2 + = 1 Major axis is horizontal 푎2 푏2 or 푥 − ℎ 2 푦 − 푘 2 + = 1 Major axis is vertical 푏2 푎2 The foci lie on the major axis, 푐 units from the center, with 푐2 = 푎2 − 푏2 The ration 푒 = 푐/푎 is called the eccentricity of an ellipse. Example 3. Find the center, vertices, and foci of the ellipse 4푥2 + 푦2 − 8푥 + 4푦 − 8 = 0. Lecture Note 10.1 Conics and Calculus Applications of Ellipse Movements of Planets in Our Solar System The planets in our solar system move in elliptical orbits, with the Sun at a focus. Earth satellites also travel in elliptic orbits, with Earth at a focus. Lecture Note 10.1 Conics and Calculus Applications of Ellipse Theorem 10.4 Reflection Property of an Ellipse Let 푃 be a point on an ellipse. The tangent line to the ellipse at point 푃 makes equal angles with the lines through 푃 and the foci. Lecture Note 10.1 Conics and Calculus Example Find the area of the ellipse 푥2 푦2 + = 1. 푎2 푏2 푥2 푦2 Example 5. Show that the circumference of the ellipse + = 1 is 푎2 푏2 휋 2 푐 4푎 1 − 푒2 sin2 휃 푑휃, where 푒 = . 0 푎 Elliptic Integral Example 6. Use the elliptic integral in Example 5 and Simpson’s Rule with 푛 = 4 to approximate the circumference of the ellipse 푥2 푦2 + = 1 25 16 Lecture Note 10.1 Conics and Calculus Hyperbolas An hyperbola is the set of points in a plane the difference of whose distances from two fixed points 퐹1 and 퐹2 is a constant. These two fixed points are called the foci. The peaks are called the vertices of the hyperbola. 푏 푏 An hyperbola has two slant asymptotes: 푦 = 푥 and 푦 = − 푥 푎 푎 푏 푦 = 푥 푎 푃(푥, 푦) Standard Equation Let the difference of the distances from a point on the ellipse to the ±2푎 (0,0) foci be . Then 2 2 (−푐, 0)퐹1 퐹2(푐, 0) 푥 푦 − = 1 푎2 푏2 where 푐2 = 푎2 + 푏2. vertices 푏 (−푎, 0) and (푎, 0) 푦 = − 푥 푎 Lecture Note 10.1 Conics and Calculus 푏 푦 = 푥 푎 Standard Equation 푃(푥, 푦) Let the difference of the distances from a point on the ellipse to the foci be ±2푎. Then (0,0) 2 2 (−푐, 0)퐹 퐹 (푐, 0) 푥 푦 1 2 − = 1 푎2 푏2 where 푐2 = 푎2 + 푏2. vertices 푦 = ± 푏/푎 푥. 푏 Asymptotes: (−푎, 0) and (푎, 0) 푦 = − 푥 푎 푎 푃(푥, 푦) 푦 = 푥 푏 Standard Equation (0, 푐) Let the difference of the distances 퐹2 from a point on the ellipse to the vertices (0, 푎) and (0, −푎) foci be ±2푎.
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