CONIC SECTIONS. Teacher's Guide

Transcription

1 CONIC SECTIONS Teacher's Guide

2 This guide is designed for use with Conic Sections, a series of three programs produced by TVOntario, the television service of the Ontario Educational Communications Authority. The series is available on videotape to educational institutions and nonprofit organizations. The series Producer/Director: David Chamberlain Project Officers: John Amadio David Way The guide Project Leader: David Chamberlain Author: Ron Carr Editor: Merlin Cheung Designer: Marie-Jos6e Bisson Ordering Information To order this publicaiton or videotapes of the programs in the Conic Sections series, or for additional information, please contact the following: Ontario TVOntario Sales and Licensing Box 200, Station Q Toronto, Ontario M4T 2T1 (416) United States TVOntario U.S. Sales Office 901 Kildaire Farm Road Building A Cary, North Carolina Phone: Fax: ussales@tvo.or g Note: These tapes are available in VHS, Betarnax I,11, and III, and 3/4" formats. Please specify format when ordering. Program BPN Slicing the Cone Circles The Ellipse The Parabola The Hyperbola Conic Reflections Copyright 1990 by The Ontario Educational Communications Authority. All rights reserved. Printed!n Canada 3667/90

3 CONIC SECTIONS Introduction iii Slicing the Cone 1 Circles 3 The Ellipse 6 The Parabola 9 The Hyperbola 11 Conic Reflections 1 3

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5 I NTRODUCTION CONIC SECTIONS Applications of the properties of curves called conics are evident in our everyday life: in satellite dishes, reflectors for flashlights and automobile headlights, the paths of thrown or projected objects, the paths of satellites and planets, and arches of bridges. Conks can be defined in many different ways. Originally, the circle, parabola, ellipse, and hyperbola were obtained by the Greek mathematician Apollonius (third century B.C.) by considering sections or slices of a right-circular cone. These curves are found by varying the angle of the slicing plane. This six-program Conic Sections series examines the conics; considers them on the Cartesian Plane; and applies simple transformations to "move" them from standard position to other positions on the plane. Methods of construction are also introduced. In the last program, basic applications of conics are illustrated and discussed. Conic Sections is another series in Concepts in Mathematics which uses contemporary three-dimensional animation techniques. iii

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7 OBJECTIVES After viewing this program and completing several activities and problems, students should be able to do the following: 1. Understand the relationship between the conics (the circle, ellipse, parabola, and hyperbola) and "slices" of a right-circular cone. 2. Define the circle, ellipse, parabola, and hyperbola using locus-type definitions. 3. Using paper, pencil, and string, construct the conics. 4. Understand the concept of "degenerate conic" as related to the slicing of a cone. SLICING THE CONE PREVIEWING ACTIVITIES 1. Before viewing the first program, students should review their knowledge of the circle and related terms such as centre, radius, diameter, sector, tangent, secant, and arc. 2. Find the circumference and area of each of the following circles: 3. Find the length of an arc of a circle with radius 25 cm and subtended by an angle at the centre equal to the following measures: (i) 90 (ii) 45 (iii) Students should be asked to bring to class a string, pencil, ruler, two thumb tacks, and a piece of cardboard on which to draw. Using one tack, the string, and the ruler, construct the following circles: (i) radius 15 cm (ii) diameter 40 cm 5. Introduce a right-circular cone to the class. Discuss the properties of this cone including the circular base and the fact that the line from the vertex to the centre of the base is perpendicular to the base (height). 1

8 PROGRAM DESCRIPTION SLICING T NE The first program begins with the introduction of the space travellers, Ed and Charlie, whom we have met in other Concepts in Mathematics series. Their spacecraft has crashed on an island which is unknown to them. They quickly realize that they have experienced time-travel to hundreds of years in the past and, spotting an ancient Greek temple, they go to "check it out." Using a revolving double-napped right-circular cone, the conic sections are introduced. By slicing the cone parallel to its base, a circle is obtained. By varying the angle of the slice, the other conics (parabola, ellipse, and hyperbola) are presented. The degenerate conics consisting of a point, a line, or two intersecting lines, are also illustrated using the cone. The program continues with demonstrations of how conics can be drawn on paper using a pencil, string and tacks. Each of the curves is constructed and locus definitions of the conics are discussed. Ed and Charlie feel that this new information obtained from the ancient Greeks might just prove to be useful to them and they begin their task of repairing their ship and attempting to return to the twentieth century. POSTVIEWING ACTIVITIES 1. In your notebook, attempt to draw a right-circular cone and draw "slices" that will illustrate a circle, ellipse, and parabola. Using a separate diagram involving a double-napped cone, attempt to illustrate a hyperbola. 2. Write the definition of a circle. 3. (a) Write a definition of an ellipse. (b) The two fixed points are called foci (plural of "focus"), and the distance from a focus point to the curve is sometimes called a focal radius. Write a definition of an ellipse using the term "focal radii." (c) Using your cardboard, pencil, string, and tacks, draw the ellipse with the sum of its focal radii equal to 20 cm and the distance between the tacks equal to 8 cm. 4. Write the definition of a parabola. Use the terms "directrix" for the fixed line and "focus point" for the fixed point. 5. In a hyperbola, the two fixed points are also called foci and the distance from a focus point to the curve is called a focal radius. Write a definition of a hyperbola using the term "foci" and the phrase "difference of the focal radii." 2

9 OBJECTIVES After viewing this program and completing several suggested activities and exercises, students will be able to do the following: 1. Using a definition, develop the equation for a circle on a Cartesian Plane, given the radius and the centre at (0, 0). 2. Find the images of points under given translations. 3. Find the equation of the image of a circle, centred at the origin, under a given translation. 4. Find the equation of a circle, centre (h, k), and radius r units. 5. Find the centre and radius of a circle by completing the square and analyzing the resulting equation. CIRCLES PREVIEWING ACTIVITIES 1. In this and subsequent programs, the Cartesian Plane is used, points are plotted, and curves are sketched. The teacher should review this system with the class. 2. Locate the following points (ordered pairs) on a Cartesian Plane: (3, 2), (5, 0), (-3, 4), (6, -1), (-2, -4). 3. Construct a table of values and sketch the graph of the relation defined by y = 2x - 3. What are the X and Y intercepts of this line? 4. On one coordinate plane, sketch the graphs of the lines defined by the following equations: y = 4x - 2 and 3x + 4y = 11. At what point do these lines intersect? 5. Review the definition of a circle. 6. Review the Pythagorean Relationship relating the lengths of the sides of a right triangle. Find the length of the hypotenuse of the right triangle given 6 cm and 8 cm as the lengths of the other two sides. 7. Plot the points A(-2,1) and B(4, 5) on a Cartesian Plane. Draw a horizontal line through A and a vertical line through B, meeting at C. What are the coordinates of C? Find the lengths of the line segment AC and the line segment BC. Using the Pythagorean Relationship, calculate the length of AB. 8. Repeat activity 7 using the points P, (x 1, y) and P2 (x 2, Find the y2). length of Px P2. This is an expression for the distance between two points. 9. Transformations are used in this program. Find the images of the points D(1, 3) and E(3, -5) under the translation defined by (x, y) --> (x+2, y+4). 10. Find the image of the point F(3, -1) under the following stretches: (i) (x, y) -~ (3x, y) (iii) (x, y) -~ (-3x, 2y) (ii) (x, y) --~. (x, 6y) (iv) (x, y) --~ ( 3 x, y) 3

10 PROGRAM DESCRIPTION CIRCLES This program opens with the standard locus definition of a circle, with "centre" and "radius" defined. Using an application of the Pythagorean Relationship to find the distance between two points on the Cartesian Plane, the equation of a circle centred at (0, 0) is constructed. The general equation of a circle in this position is also discussed. The concept of transformations in general is introduced, and a translation, in particular, is applied to find the equation of a circle which is not centred at the origin. Ed and Charlie continue rebuilding their spacecraft using circular shapes, but are puzzled about the odd-shaped elliptical-shaped objects which they have encountered. POSTVIEWING ACTIVITIES 1. By comparing each equation with the general equation of a circle, x2 + y2 = rz, state the centre and radius of the following: (i) x2 + y2 = 81 (ii) (iii) x2 + y2-4 = 0 3x2 + 3y 2 = Sketch the graphs of the circles in activity Write the equations of the following circles: (i) centre (0, 0), radius 5 units (ii) centre (0, 0), radius 8.5 units 4. Find the equation of the image of the line defined by 2x + 3y = 6 under a (2,-5) translation. Do this question using two methods: (i) find two points on the original line, find the images of these points under the translation, and write the equation of the line through these two points, and (ii) set up a second coordinate system, - let (x, y)~(x + 2, y - 5) = (u, v), solve this system for x and y, and substitute the expressions in u and v into the equation. 5. Find the image of the circle defined by x 2 + y2 = 4 under the translation defined by (x, y) -3 (x - 5, y + 6). Sketch the graphs of both circles on one graph. 6. Repeat activity 5 for the following circles and translations: (i) x 2 + y2 =1, (3, 2) translation (ii) x2 + y2 = 9, (0, 4) translation 7. State the centre and radius of each of the following circles: (i) (x - 2)2 + (y - 5)2 = 49 (ii) (x + 4)2 + (y )2 = 4 4

11 8. Find the equations of the circles, given the following graphs: CIRCLES 9. By completing the squares, find the centres and radii of the circles defined by the following equations: (i) x2-2x + y2-6y = -9 (ii) x2 + y2 + 4x + 10y - 7 = Find the image of the circle defined by x2 + y2 = 9 under the stretch defined by (x, y) --$ (2x, y). Using a table of values, sketch the graph of the circle and the image on one coordinate system. Is the image a circle? What is the name of this type of curve? 5

12 T ELLIPSE OBJECTIVES After viewing this program and completing several suggested activities and exercises, students will be able to do the following: Find the images of points under given dilatations (dilations) and stretches. 2. Find the equation of an ellipse, centred at (0, 0), by applying a stretch parallel to a coordinate axis to a circle centred at the origin. 3. Find the equations of ellipses obtained by applying two-way stretches to circles. 4. Write equations of ellipses in standard position, and find the lengths of the major and minor axes by writing equations in standard form. 5. Find the equations of images of ellipses under given translations. 6. By completing the squares, determine the centre and lengths of the major and minor axes of ellipses which are not in standard position. PREVIEWING ACTIVITIES Sketch the graphs of the following circles: (i) x2+ y2 = 49 (ii) 2x 2 + 2y2 = 8 2. By completing the squares, find the centres and radii of the following circles: (i) x2 +y2 +6x+6y=7 40 3x2 + 3y2-12x + 6y- 12 = 0 3. Find the image of the line defined by 3x - 5y = 15 under the following transformations: (1) a (3, -6) translation (ii) a dilatation of factor 3 (iii) a stretch parallel to the Y-axis of factor 2 (iv) a stretch parallel to the X-axis of factor 0.5 (v) the two-way stretch (x, y) -> (3x, 2y) 4. Review the "sum of the focal radii" definition of an ellipse. PROGRAM DESCRIPTION While Ed and Charlie are questioning the need for ellipses, as opposed to "perfect circles," the elliptical orbits in our solar system are briefly presented. As well, the action of slicing a cone to obtain elliptical shapes in Program One is reviewed. An ellipse is constructed on a coordinate plane by stretching a circle using another transformation called a dilation. First, the circle is stretched vertically, then stretched horizontally and, finally, in both directions at once. The general equations of an ellipse in standard position (centred at (0, 0) and with axes on the coordinate axes) are discussed. 6

13 Ed and Charlie are amazed at their special powers which allow them to produce conics in the temple. PQSTVIEWING ACTIVITIES 1. Find the image of the circle x2 + y 2 =1 under the transformation (x, y) (4x, y). Sketch the graphs of the original circle and the image ellipse. 2. Find the image of the circle x 2 + y2 = 1 under the two-way stretch (x, y) --~ (2x, 3y). Sketch the graph of the image curve. 3. State the transformation which will produce the given ellipses from an original unit circle, centred at the origin: THE ELLIPSE 4. Write the following equations of ellipses in standard form: (i) 4x2 + 9y2 = 36 (ii) 9y2 + x2 = 9 (iii) 3X2 + 4y2 = 24 (iv) 5x 2 + 2y2 = 1 5. (a) By analyzing the equations obtained in activity 4, state the values of a and b for each ellipse. (The usual convention is to choose a to be the larger of the two constants, and b as the smaller of the two.) (b) The length of the major axis of an ellipse is 2a units and the length of the minor axis is 2b units. The constants a and b are al- 7

14 T ELLIPSE ways considered to be positive. State the lengths of the major and minor axes of the ellipses in activity The vertices of an ellipse are Vl (a, 0) and V2 (-a, 0) for an ellipse with the major axis on the X-axis ( (0, a), and (0, -a) for an ellipse with the major axis on the Y-axis. State the coordinates of the vertices of the ellipses in activity The ellipse with the general standard equation ~ + =1 has its focus points (foci) on the X-axis. These points are F l (c, 0) and F2 (-c, 0) where c is calculated using the relationship a 2 = b2 + c2. Find the foci of the ellipse defined by x2 + i,2 16 =1. 8. The same relationship as in activity 7 holds for ellipses with major axes on the Y-axis. Find the foci of the ellipse defined by 4x 2 + 3y2 = Write the equations of the following ellipses in standard position: 10. Find the image of the ellipse 4x 2 + 9y2 = 36 under the translation (x, y) --> (x + 3, y - 2). What is the centre of the image ellipse? What are the vertices? 11. By completing the squares, find the "centre" of the ellipse defined by 3x2 + 5y'- 6x - 20y + 8 = 0. Write this resulting equation in "standard form." What are the lengths of the major and minor axes? 8

15 OBJECTIVES After viewing this program and completing several suggested activities and problems, students will be able to do the following: 1. Sketch the graph of the parabola defined by y = x 2, describe its symmetry, and locate its vertex. 2. Find the image of y = x2 under given translations. 3. Find the image of y = x2 under given stretches. 4. By completing the square, find the vertex and axis of symmetry of a parabola in the form y = axe + bx + c. 5. Understand the roles and implications of varying values of a, h and k in the equation y = a(x - h)2 + k. THE PARABOLA 6. Solve word problems related to bridges and arches. PREVIEWING ACTIVITIES 1. Review the focus-directrix definition of a parabola. 2. Sketch the graphs of the following curves: (i) x2 + y2 = 25 (ii) 9x2 + y2 = 9 (iii) y = x2 3. Discuss the different meanings of -x2 and (-x) Sketch the graphs of the following on the same set of coordinate axes: (i) y = x2 (iii) x = y2 (ii) y = -x2 (iv) x = -y2 5. Find the image of the ellipse defined by 4x2 + 9y2 = 36 under the stretch (x,y) -a (3x, 0.5y). PROGRAM DESCRIPTION The parabola is introduced using arched bridges as examples, and the slice of a cone'which produces a parabola is reviewed. The parabola with vertex at (0, 0) and with axis of symmetry on the Y-axis is presented and its basic equation, y = x 2, is discussed. Applying reflections and translations, the equations of parabolas in other positions on the coordinate plane are derived. Using stretches, the shape of the parabola is changed. The general equation of a parabola with vertical symmetry, y = a(x - h)2 + k is discussed and the roles of a, h, and k are examined. Using the Toronto Skydome as an example, the equation of a parabolic roof is determined (by placing the X and Y axes in advantageous positions). Ed and Charlie are very impressed with this solution and decide that a retractable roof for their spacecraft may be just what is needed. 9

16 POSTVIEWING ACTIVITIES T 1 0 RABO 1. Find the equations of the images of the parabola defined by y = x2 under the following transformations: (i) (x, y) (x, -y) (ii) (x,y)~(x+2,y+1) (iii) (x, y) (x - 4, y + 3) (iv) (x, y) (x, 3y) (v) (x, y) -~ (4x, y) 2. Sketch the graphs of the original and image parabolas in activity 1. Use one set of coordinate axes. 3. State the vertices of the parabolas defined by the following: (i) y = x2 (ii) y = (x - 2)2 (iii) y = -4 (x + 1)2 4. Sketch the graphs of y = 2x 2 and y = -2x2 on the same set of axes. Comparing the equations to the equation y = axe, if a is positive, then the parabola opens upward. If a is negative, the parabola opens downward. 5. By completing the square, find the coordinates of the vertex and the equation of the axis of symmetry for each of the following parabolas. Sketch the graph of each. (i) y = x 2-4x + 4 (iii) y = -x2 + 6x + 1 (ii) y = 2x2 + 4x + 4 (iv) y = 4x2-24x A bridge over a small river is in the form of a parabolic arch. The width of the arch at the water level is 25 m and the height of the arch at its highest point is 40 m. By considering coordinate axes intersecting at the vertex of the parabola, find the equation of the curve. 7. A cable of a suspension bridge hangs in the shape of a parabola. The heights of the supporting towers above the water level are both 100 m, and the towers are 120 m apart. The roadway is 10 m above the water. By choosing appropriate axes, find an equation for the parabola. 120m 8. In activity 7, find the length of a "hanger cable" which is 15 m from the vertex of the cable. 10. Sketch the graphs of the parabolas defined by y = 2x 2 and y = 4x2. Describe the effect of increasing or decreasing the value of a on the shape of a parabola. Ty

17 OBJECTIVES After viewing this program and completing several suggested exercises and problems, students will be able to do the following: 1. Recognize the equation of a hyperbola in standard position. 2. Sketch the graph of the hyperbola defined by y2 - x2 = Understand the concept of an asymptote and how to use the asymptotes to sketch a given hyperbola. 4. Find the equations of the asymptotes of a hyperbola in standard position. 5. Write the equation in standard form of a hyperbola in standard position. 6. Determine the lengths of the transverse and conjugate axes. 7. Apply transformations to find the images of given hyperbolas. THE HYPERBOLA PREVIEWING ACTIVITIES 1. Sketch the graphs of the following curves: (i) 2x2 + 2y2 = 16 (ii) x2 + 4y2-16 = 0 (iii) y = x2-2x-2 2. Find the equation of the image of 9x 2 + y2 = 9 under a (-3, 4) translation. 3. Construct a table of values and sketch the graph of y = z for x > 0. As x increases, y gets closer and closer to the X-axis. Use this example to briefly introduce the concept of an asymptote. 4. Find the coordinates of the vertices and foci, and the lengths of the major and minor axes of the following ellipses: (i) 4x 2 + 9y2 = 36 (ii) 16x2 + y2 = Review with the class the "slice" of a double-napped cone which will produce a hyperbola. PROGRAM DESCRIPTION Ed and Charlie are still interested in learning about conics and decide to ask about the hyperbola. When examples are presented, there is some discussion as to the apparent similarity between the shapes of the parabola and one branch of a hyperbola. Using an example on the Cartesian Plane, the concept of an asymptote is introduced to show that the two shapes are actually very different. Terminology related to the hyperbola is introduced and explained. Vertices, and the transverse and conjugate axes are illustrated. Using a reflection in the line y = x, the two standard positions of the hyperbola are discussed. 1 1

18 T PE BO POSTVIEWING ACTIVITIES 1. State the "difference of the focal radii" definition of a hyperbola. 2. Construct a table of values and sketch the graph of x 2 - y2 = The equations of the asymptotes of a hyperbola can be obtained by setting the constant term equal to zero, and solving the resulting equation to obtain the equations of two lines. Find the equations of the asymptotes of the following hyperbolas: (i) 4x2- y2 = 4 (ii) 9x2-16y2 = -144 (iii) x2 - y2= The vertices of a hyperbola can be found at the points where the curves cross one of the coordinate axes. Find the vertices of the hyperbolas in activity Sketch the graphs of the hyperbolas in activity 3 by sketching the graphs of the asymptotes, locating the vertices, and sketching smooth curves. 6. Write the following equations in standard form: (i) 9x2 - y2 = 9 (ii) 9x2 - y 2 = -9 (iii) 4x2-9y2 = -36 (iv) 6x2 - y2= 1 7. The length of the transverse axis of the hyperbola defined by a2 - = 1 is 2a units and the length of the conjugate axis is 2b units. State the lengths of the axes of the hyperbola defined by x A hyperbola with its vertices on the Y-axis has an equation in the form b2 - a2 = -1. State the vertices and lengths of the axes of the hyperbola defined by 2L 9. Find the image of the hyperbola defined by 3x 2-4y 2 = 12 under the translation (x, y) -~ (x + 3, y -1). Sketch the graph of the image. 1 2

19 OBJECTIVES After viewing this program, students will be able to do the following: 1. Understand some of the practical uses and applications of the properties of conic shapes in everyday life. 2. Understand the concept of eccentricity as related to the determination of various conics. PREVIEWING ACTIVITIES Before viewing this program, the teacher should ask members of the class to bring in, if possible, flashlights, automobile headlights, pamphlets describing television satellite receivers, information regarding sonic booms, etc. CONIC REFLECTIONS PROGRAM DESCRIPTION This program opens with a brief discussion of the eccentricity of a conic, how various values produce different conics, and how the shape of an individual conic is influenced by a change in the value of the eccentricity. Ed and Charlie look for some practical uses for the four conics, and the reflector properties of the conics are discussed. In a circle, sound produced at the centre is reflected back to the centre. Sound produced at one focus of an ellipse will reflect off the ellipse to the other focus. Parallel rays of light aimed at a parabolic surface will all reflect through the focus of the parabola. The practical use of this property of parabolas is illustrated by the use of dish satellite antennas, the mirrors in telescopes, and the automobile headlight. Properties of the hyperbola are discussed using sonic booms of jet aircraft, and interplanetary flight paths. Ed and Charlie finally are ready to resume their journey, using a circular flight deck, ellipsoidal headlights, parabolic antennas, and a hyperbolic flight trajectory. POSTVIEWING ACTIVITIES 1. This series has defined the parabola using a focus-directrix definition. The ellipse and hyperbola can also be defined using foci and directrices. This may be an opportunity to discuss these definitions with the class to gain a further understanding of the eccentricity of an ellipse. 1 3

20 2. Using the definitions discussed in activity 1, find the equations the following conics: (i) F(4, 0), directrix x = -4, eccentricity = 1 (ii) F(3, 0), directrix x =, eccentricity =i (iii) F(0, 5), directrix y =, eccentricity = 4 CONIC REFLECTIONS 3. Use the items brought in by students to illustrate the practical uses of the reflector properties of the ellipse and parabola. 1 4