Topics in Analytic Geometry

Transcription

1 Topics in Analtic Geometr. Lines. Introduction to Conics: Parabolas. Ellipses. Hperbolas.5 Rotation of Conics. Parametric Equations.7 Polar Coordinates.8 Graphs of Polar Equations.9 Polar Equations of Conics In Mathematics A conic is a collection of points satisfing a geometric propert. In Real Life Conics are used as models in construction, planetar orbits, radio navigation, and projectile motion. For instance, ou can use conics to model the orbits of the planets as the move about the sun. Using the techniques presented in this chapter, ou can determine the distances between the planets and the center of the sun. (See Eercises 55, page 79.) Mike Agliolo/Photo Researchers, Inc. IN CAREERS There are man careers that use conics and other topics in analtic geometr. Several are listed below. Home Contractor Eercise 9, page 7 Civil Engineer Eercises 7 and 7, page 7 Artist Eercise 5, page 759 Astronomer Eercises and, page 79 75

2 7 Chapter Topics in Analtic Geometr. LINES What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it The inclination of a line can be used to measure heights indirectl. For instance, in Eercise 7 on page 7, the inclination of a line can be used to determine the change in elevation from the base to the top of the Falls Incline Railwa in Niagara Falls, Ontario, Canada. Inclination of a Line In Section., ou learned that the graph of the linear equation m b is a nonvertical line with slope m and -intercept, b. There, the slope of a line was described as the rate of change in with respect to. In this section, ou will look at the slope of a line in terms of the angle of inclination of the line. Ever nonhorizontal line must intersect the -ais. The angle formed b such an intersection determines the inclination of the line, as specified in the following definition. Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line. (See Figure..) θ = JTB Photo/Japan Travel Bureau/PhotoLibrar θ = Horizontal Line Vertical Line Acute Angle Obtuse Angle FIGURE. The inclination of a line is related to its slope in the following manner. θ θ Inclination and Slope If a nonvertical line has inclination and slope m, then m tan. For a proof of this relation between inclination and slope, see Proofs in Mathematics on page 8.

3 Section. Lines 77 Eample Finding the Inclination of a Line FIGURE. = θ = 5 Find the inclination of the line. Solution The slope of this line is m. So, its inclination is determined from the equation tan. From Figure., it follows that < arctan. The angle of inclination is radian or 5. Now tr Eercise 7. < This means that. Eample Finding the Inclination of a Line Find the inclination of the line. Solution The slope of this line is m. So, its inclination is determined from the equation FIGURE. + = θ. θ = θ θ tan. From Figure., it follows that < arctan <. This means that The angle of inclination is about.55 radians or about.. Now tr Eercise. FIGURE. θ θ θ The Angle Between Two Lines Two distinct lines in a plane are either parallel or intersecting. If the intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure., ou can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations and, where < and <, the angle between the two lines is.

4 78 Chapter Topics in Analtic Geometr You can use the formula for the tangent of the difference of two angles tan tan tan tan tan tan to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan m m m m. Eample Finding the Angle Between Two Lines FIGURE.5 + = θ 79.7 = Find the angle between the two lines. Line : Line : Solution The two lines have slopes of m and m, respectivel. So, the tangent of the angle between the two lines is tan m m m. m Finall, ou can conclude that the angle is arctan as shown in Figure.5..9 radians 79.7 Now tr Eercise. The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure.. (, ) (, ) d Distance Between a Point and a Line The distance between the point, and the line A B C is C d A B. A B FIGURE. Remember that the values of A, B, and C in this distance formula correspond to the general equation of a line, A B C. For a proof of this formula for the distance between a point and a line, see Proofs in Mathematics on page 8.

5 Section. Lines 79 = + (, ) 5 FIGURE.7 Eample Finding the Distance Between a Point and a Line Find the distance between the point, and the line. Solution The general form of the equation is. So, the distance between the point and the line is d 8.58 units. 5 The line and the point are shown in Figure.7. Now tr Eercise 5. Eample 5 An Application of Two Distance Formulas 5 B (, ) h C (5, ) A (, ) 5 FIGURE.8 Figure.8 shows a triangle with vertices A,, B,, and C5,. a. Find the altitude h from verte B to side AC. b. Find the area of the triangle. Solution a. To find the altitude, use the formula for the distance between line AC and the point,. The equation of line AC is obtained as follows. Slope: Equation: m 5 8 Point-slope form Multipl each side b. General form So, the distance between this line and the point, is Altitude h 7 units. b. Using the formula for the distance between two points, ou can find the length of the base AC to be b units. Finall, the area of the triangle in Figure.8 is Distance Formula Simplif. Simplif. A bh 7 7 square units. Now tr Eercise 59. Formula for the area of a triangle Substitute for b and h. Simplif.

6 7 Chapter Topics in Analtic Geometr. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. The of a nonhorizontal line is the positive angle (less than ) measured counterclockwise from the -ais to the line.. If a nonvertical line has inclination and slope m, then m.. If two nonperpendicular lines have slopes m and m, the angle between the two lines is tan.. The distance between the point, and the line A B C is given b d. SKILLS AND APPLICATIONS In Eercises 5, find the slope of the line with inclination radians. radians. radians. radians In Eercises 8, find the inclination degrees) of the line with a slope of m m m m m m m 5 θ = θ = θ = θ = (in radians and In Eercises 7, find the inclination degrees) of the line ,,,,,,,,, 8, 8,,,,,,, 5, In Eercises 7, find the angle between the lines. θ (in radians and (in radians and degrees) θ In Eercises 9, find the inclination (in radians and degrees) of the line passing through the points. 9..,,,,,,

7 Section. Lines θ θ Point 5., 5., 55., 5., 57., 8 58., Line In Eercises 59, the points represent the vertices of a triangle. (a) Draw triangle ABC in the coordinate plane, (b) find the altitude from verte B of the triangle to side AC, and (c) find the area of the triangle A,, B,, C, A,, B, 5, C 5, A,, B,, A, 5, B,, C, In Eercises and, find the distance between the parallel lines... 5 C 5, ANGLE MEASUREMENT In Eercises 7 5, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles (, 8) (, 5) (, 5) 8 (, ) (, ) (, ) 5. ROAD GRADE A straight road rises with an inclination of. radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road (, ) (, ) (, ) (, ) (, ) (, ) mi. radian In Eercises 5 58, find the distance between the point and the line. Point 5., 5., Line. ROAD GRADE A straight road rises with an inclination of. radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

8 7 Chapter Topics in Analtic Geometr 7. PITCH OF A ROOF A roof has a rise of feet for ever horizontal change of 5 feet (see figure). Find the inclination of the roof. 8. CONVEYOR DESIGN A moving conveor is built so that it rises meter for each meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveor. (c) The conveor runs between two floors in a factor. The distance between the floors is 5 meters. Find the length of the conveor. 9. TRUSS Find the angles and shown in the drawing of the roof truss. 7. The Falls Incline Railwa in Niagara Falls, Ontario, Canada is an inclined railwa that was designed to carr people from the Cit of Niagara Falls to Queen Victoria Park. The railwa is approimatel 7 feet long with a % uphill grade (see figure). θ 5 ft ft ft 7 ft α β 9 ft Not drawn to scale ft ft (c) Using the origin of a rectangular coordinate sstem as the base of the inclined plane, find the equation of the line that models the railwa track. (d) Sketch a graph of the equation ou found in part (c). EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. A line that has an inclination greater than radians has a negative slope. 7. To find the angle between two lines whose angles of inclination and are known, substitute and for m and m, respectivel, in the formula for the angle between two lines. 7. Consider a line with slope m and -intercept,. (a) Write the distance d between the origin and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the origin and the line. (d) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. 7. CAPSTONE Discuss wh the inclination of a line can be an angle that is larger than, but the angle between two lines cannot be larger than. Decide whether the following statement is true or false: The inclination of a line is the angle between the line and the -ais. Eplain. 75. Consider a line with slope m and -intercept,. (a) Write the distance d between the point, and the line as a function of m. (b) Graph the function in part (a). (c) Find the slope that ields the maimum distance between the point and the line. (d) Is it possible for the distance to be? If so, what is the slope of the line that ields a distance of? (e) Find the asmptote of the graph in part (b) and interpret its meaning in the contet of the problem. (a) Find the inclination of the railwa. (b) Find the change in elevation from the base to the top of the railwa.

9 Section. Introduction to Conics: Parabolas 7. INTRODUCTION TO CONICS: PARABOLAS What ou should learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of parabolas in standard form and graph parabolas. Use the reflective propert of parabolas to solve real-life problems. Wh ou should learn it Parabolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 7 on page 79, a parabola is used to model the cables of the Golden Gate Bridge. Conics Conic sections were discovered during the classical Greek period, to B.C. The earl Greeks were concerned largel with the geometric properties of conics. It was not until the 7th centur that the broad applicabilit of conics became apparent and plaed a prominent role in the earl development of calculus. A conic section (or simpl conic) is the intersection of a plane and a doublenapped cone. Notice in Figure.9 that in the formation of the four basic conics, the intersecting plane does not pass through the verte of the cone. When the plane does pass through the verte, the resulting figure is a degenerate conic, as shown in Figure.. Cosmo Condina/The Image Bank/ Gett Images Circle Ellipse Parabola Hperbola FIGURE.9 Basic Conics Point Line Two Intersecting FIGURE. Degenerate Conics Lines There are several was to approach the stud of conics. You could begin b defining conics in terms of the intersections of planes and cones, as the Greeks did, or ou could define them algebraicall, in terms of the general second-degree equation A B C D E F. However, ou will stud a third approach, in which each of the conics is defined as a locus (collection) of points satisfing a geometric propert. For eample, in Section., ou learned that a circle is defined as the collection of all points, that are equidistant from a fied point h, k. This leads to the standard form of the equation of a circle h k r. Equation of circle

10 7 Chapter Topics in Analtic Geometr Parabolas In Section., ou learned that the graph of the quadratic function f a b c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. Definition of Parabola A parabola is the set of all points, in a plane that are equidistant from a fied line (directri) and a fied point (focus) not on the line. Directri FIGURE. Focus Verte Parabola d d d d The midpoint between the focus and the directri is called the verte, and the line passing through the focus and the verte is called the ais of the parabola. Note in Figure. that a parabola is smmetric with respect to its ais. Using the definition of a parabola, ou can derive the following standard form of the equation of a parabola whose directri is parallel to the -ais or to the -ais. Standard Equation of a Parabola The standard form of the equation of a parabola with verte at h, k is as follows. h p k, p k p h, p Vertical ais, directri: k p Horizontal ais, directri: h p The focus lies on the ais p units (directed distance) from the verte. If the verte is at the origin,, the equation takes one of the following forms. p p See Figure.. Vertical ais Horizontal ais For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 85. p > Verte: ( h, k) Ais: = h Focus: ( h, k + p) Directri: = k p p < Ais: = h Directri: = k p Verte: (h, k) Focus: (h, k + p) Directri: = h p p > Verte: ( h, k) Focus: ( h + p, k) Ais: = k Focus: (h + p, k) Directri: = h p p < Verte: (h, k) Ais: = k (a) h p k Vertical ais: p > FIGURE. (b) h p k Vertical ais: p < (c) k p h Horizontal ais: p > (d) k p h Horizontal ais: p <

11 Section. Introduction to Conics: Parabolas 75 TECHNOLOGY Use a graphing utilit to confirm the equation found in Eample. In order to graph the equation, ou ma have to use two separate equations: and 8 8. Upper part Lower part Eample Verte at the Origin Find the standard equation of the parabola with verte at the origin and focus,. Solution The ais of the parabola is horizontal, passing through, and,, as shown in Figure.. Verte = 8 Focus (, ) (, ) FIGURE. The standard form is p, where h, k, and p. So, the equation is 8. Now tr Eercise. Eample Finding the Focus of a Parabola The technique of completing the square is used to write the equation in Eample in standard form. You can review completing the square in Appendi A.5. Verte (, ) Focus (, ) = + FIGURE. Find the focus of the parabola given b. Solution To find the focus, convert to standard form b completing the square. Write original equation. Multipl each side b. Add to each side. Complete the square. Combine like terms. Standard form Comparing this equation with h p k ou can conclude that h, k, and p. Because p is negative, the parabola opens downward, as shown in Figure.. So, the focus of the parabola is h, k p,. Now tr Eercise.

12 7 Chapter Topics in Analtic Geometr Eample Finding the Standard Equation of a Parabola 8 FIGURE.5 Light source at focus FIGURE. Focus Focus ( ) = ( ) Focus (, ) Verte (, ) 8 Ais Parabolic reflector: Light is reflected in parallel ras. α Ais P Find the standard form of the equation of the parabola with verte, and focus,. Then write the quadratic form of the equation. Solution Because the ais of the parabola is vertical, passing through, consider the equation h p k where h, k, and p. So, the standard form is. You can obtain the more common quadratic form as follows. Write original equation. Multipl. Add to each side. The graph of this parabola is shown in Figure.5. Application Now tr Eercise 55. Divide each side b. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The specific focal chord perpendicular to the ais of the parabola is called the latus rectum. Parabolas occur in a wide variet of applications. For instance, a parabolic reflector can be formed b revolving a parabola around its ais. The resulting surface has the propert that all incoming ras parallel to the ais are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversel, the light ras emanating from the focus of a parabolic reflector used in a flashlight are all parallel to one another, as shown in Figure.. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. and,, FIGURE.7 α Tangent line Reflective Propert of a Parabola The tangent line to a parabola at a point P makes equal angles with the following two lines (see Figure.7).. The line passing through P and the focus. The ais of the parabola

13 Section. Introduction to Conics: Parabolas 77 =, d α FIGURE.8 ( ) d α (, b) TECHNOLOGY (, ) Use a graphing utilit to confirm the result of Eample. B graphing and in the same viewing window, ou should be able to see that the line touches the parabola at the point,. Eample Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given b at the point,. Solution For this parabola, p and the focus is,, as shown in Figure.8. You can find the -intercept, b of the tangent line b equating the lengths of the two sides of the isosceles triangle shown in Figure.8: and d b d 5. Note that d rather than b b. The order of subtraction for the distance is important because the distance must be positive. Setting d d produces b 5 So, the slope of the tangent line is m b. and the equation of the tangent line in slope-intercept form is. Now tr Eercise 5. You can review techniques for writing linear equations in Section.. CLASSROOM DISCUSSION Satellite Dishes Cross sections of satellite dishes are parabolic in shape. Use the figure shown to write a paragraph eplaining wh satellite dishes are parabolic. Amplifier Dish reflector Cable to radio or TV

14 78 Chapter Topics in Analtic Geometr. EXERCISES VOCABULARY: Fill in the blanks.. A is the intersection of a plane and a double-napped cone. See for worked-out solutions to odd-numbered eercises.. When a plane passes through the verte of a double-napped cone, the intersection is a.. A collection of points satisfing a geometric propert can also be referred to as a of points.. A is defined as the set of all points, in a plane that are equidistant from a fied line, called the, and a fied point, called the, not on the line. 5. The line that passes through the focus and the verte of a parabola is called the of the parabola.. The of a parabola is the midpoint between the focus and the directri. 7. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a. 8. A line is to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. SKILLS AND APPLICATIONS In Eercises 9, describe in words how a plane could intersect with the double-napped cone shown to form the conic section. (e) (f) 9. Circle. Ellipse. Parabola. Hperbola In Eercises 8, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (b) (d) In Eercises 9, find the standard form of the equation of the parabola with the given characteristic(s) and verte at the origin. 9.. (, ) 8 (, ). Focus:,. Focus:,. Focus:,. Focus:, 5. Directri:. Directri: 7. Directri: 8. Directri: 9. Vertical ais and passes through the point,. Vertical ais and passes through the point,. Horizontal ais and passes through the point, 5. Horizontal ais and passes through the point, 8 8

15 Section. Introduction to Conics: Parabolas 79 In Eercises, find the verte, focus, and directri of the parabola, and sketch its graph In Eercises 7 5, find the verte, focus, and directri of the parabola. Use a graphing utilit to graph the parabola In Eercises 5, find the standard form of the equation of the parabola with the given characteristics (, ) (, ) (, ) 8 (, ) Verte:, ; focus:, 5. Verte:, ; focus:, 57. Verte:, ; directri: 58. Verte:, ; directri: 59. Focus:, ; directri:. Focus:, ; directri: 8 8 (, ) In Eercises and, change the equation of the parabola so that its graph matches the description.. ; upper half of parabola. ; lower half of parabola 8 (, ) (.5, ) (5, ) In Eercises and, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utilit to graph both equations in the same viewing window. Determine the coordinates of the point of tangenc... Parabola 8 Tangent Line In Eercises 5 8, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. 5.,, 8.,, 9 7.,, 8.,, 8 9. REVENUE The revenue R (in dollars) generated b the sale of units of a patio furniture set is given b R,5. 5 Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 7. REVENUE The revenue R (in dollars) generated b the sale of units of a digital camera is given b 5 5 R 5,55. 7 Use a graphing utilit to graph the function and approimate the number of sales that will maimize revenue. 7. SUSPENSION BRIDGE Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 8 meters apart. The top of each tower is 5 meters above the roadwa. The cables touch the roadwa midwa between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate sstem at the center of the roadwa. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table b finding the height of the suspension cables over the roadwa at a distance of meters from the center of the bridge. Distance, 5 5 Height,

16 7 Chapter Topics in Analtic Geometr 7. SATELLITE DISH The receiver in a parabolic satellite dish is.5 feet from the verte and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the verte is at the origin.) Receiver.5 ft 75. BEAM DEFLECTION A simpl supported beam is meters long and has a load at the center (see figure). The deflection of the beam at its center is centimeters. Assume that the shape of the deflected beam is parabolic. (a) Write an equation of the parabola. (Assume that the origin is at the center of the deflected beam.) (b) How far from the center of the beam is the deflection equal to centimeter? cm 7. ROAD DESIGN Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is feet wide is. foot higher in the center than it is on the sides (see figure). Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface. foot lower than in the middle? 7. HIGHWAY DESIGN Highwa engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highwa (see figure). Find an equation of the parabola. 8 ft Interstate (, 8) 8. ft Not drawn to scale 7. BEAM DEFLECTION Repeat Eercise 75 if the length of the beam is meters and the deflection of the beam at the center is centimeters. 77. FLUID FLOW Water is flowing from a horizontal pipe 8 feet above the ground. The falling stream of water has the shape of a parabola whose verte, 8 is at the end of the pipe (see figure). The stream of water strikes the ground at the point,. Find the equation of the path taken b the water. 8 ft FIGURE FOR 77 FIGURE FOR 78 m (, ) (, ) (, ) LATTICE ARCH A parabolic lattice arch is feet high at the verte. At a height of feet, the width of the lattice arch is feet (see figure). How wide is the lattice arch at ground level? 79. SATELLITE ORBIT A satellite in a -mile-high circular orbit around Earth has a velocit of approimatel 7,5 miles per hour. If this velocit is multiplied b, the satellite will have the minimum velocit necessar to escape Earth s gravit and it will follow a parabolic path with the center of Earth as the focus (see figure on the net page). Not drawn to scale 8 (, 8) Street

17 Section. Introduction to Conics: Parabolas 7 FIGURE FOR 79 (a) Find the escape velocit of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is miles). 8. PATH OF A SOFTBALL The path of a softball is modeled b , where the coordinates and are measured in feet, with corresponding to the position from which the ball was thrown. (a) Use a graphing utilit to graph the trajector of the softball. (b) Use the trace feature of the graphing utilit to approimate the highest point and the range of the trajector. PROJECTILE MOTION In Eercises 8 and 8, consider the path of a projectile projected horizontall with a velocit of v feet per second at a height of s feet, where the model for the path is In this model (in which air resistance is disregarded), is the height (in feet) of the projectile and is the horizontal distance (in feet) the projectile travels. 8. A ball is thrown from the top of a -foot tower with a velocit of 8 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontall before striking the ground? 8. A cargo plane is fling at an altitude of, feet and a speed of 5 miles per hour. A suppl crate is dropped from the plane. How man feet will the crate travel horizontall before it hits the ground? EXPLORATION miles v s. Circular orbit TRUE OR FALSE? In Eercises 8 and 8, determine whether the statement is true or false. Justif our answer. 8. It is possible for a parabola to intersect its directri. 8. If the verte and focus of a parabola are on a horizontal line, then the directri of the parabola is vertical. Parabolic path Not drawn to scale 85. Let, be the coordinates of a point on the parabola p. The equation of the line tangent to the parabola at the point is p. What is the slope of the tangent line? 8. CAPSTONE Eplain what each of the following equations represents, and how equations (a) and (b) are equivalent. (a) a h k, a (b) h p k, p (c) k p h, p 87. GRAPHICAL REASONING Consider the parabola p. (a) Use a graphing utilit to graph the parabola for p, p, p, and p. Describe the effect on the graph when p increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directl from the standard form of the equation of the parabola? Latus rectum (d) Eplain how the result of part (c) can be used as a sketching aid when graphing parabolas. 88. GEOMETRY The area of the shaded region in the figure is A 8 p b. = p (a) Find the area when p and b. Focus = p = b (b) Give a geometric eplanation of wh the area approaches as p approaches.

18 7 Chapter Topics in Analtic Geometr. ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems. Find eccentricities of ellipses. Wh ou should learn it Ellipses can be used to model and solve man tpes of real-life problems. For instance, in Eercise 5 on page 79, an ellipse is used to model the orbit of Halle s comet. Introduction The second tpe of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points, in a plane, the sum of whose distances from two distinct fied points (foci) is constant. See Figure.9. Focus d (, ) d Focus Major ais Verte Center Minor ais Verte Harvard College Observator/Photo Researchers, Inc. d d is constant. FIGURE.9 FIGURE. The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major ais, and its midpoint is the center of the ellipse. The chord perpendicular to the major ais at the center is the minor ais of the ellipse. See Figure.. You can visualize the definition of an ellipse b imagining two thumbtacks placed at the foci, as shown in Figure.. If the ends of a fied length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced b the pencil will be an ellipse. (, ) FIGURE. b + c b + c ( h, k) b To derive the standard form of the equation of an ellipse, consider the ellipse in Figure. with the following points: center, h, k; vertices, h ± a, k; foci, h ± c, k. Note that the center is the midpoint of the segment joining the foci. The sum of the distances from an point on the ellipse to the two foci is constant. Using a verte point, this constant sum is b + c = a b + c = a FIGURE. c a a c a c a Length of major ais or simpl the length of the major ais. Now, if ou let, be an point on the ellipse, the sum of the distances between, and the two foci must also be a.

19 Section. Ellipses 7 That is, h c k h c k a which, after epanding and regrouping, reduces to a c h a k a a c. Finall, in Figure., ou can see that b a c which implies that the equation of the ellipse is b h a k a b h a k b. You would obtain a similar equation in the derivation b starting with a vertical major ais. Both results are summarized as follows. Consider the equation of the ellipse h a k b. If ou let a b, then the equation can be rewritten as h k a which is the standard form of the equation of a circle with radius r a (see Section.). Geometricall, when a b for an ellipse, the major and minor aes are of equal length, and so the graph is a circle. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center h, k and major and minor aes of lengths a and b, respectivel, where < b < a, is h h Figure. shows both the horizontal and vertical orientations for an ellipse. a b Major ais is horizontal. Major ais is vertical. The foci lie on the major ais, c units from the center, with c a b. If the center is at the origin,, the equation takes one of the following forms. a b k b k a. Major ais is horizontal. b a Major ais is vertical. ( h) ( k) + = a b ( h) ( k) + = b a (h, k) b (h, k) a a b Major ais is horizontal. FIGURE. Major ais is vertical.

20 7 Chapter Topics in Analtic Geometr Eample Finding the Standard Equation of an Ellipse FIGURE. (, ) (, ) (, ) a = b = 5 Find the standard form of the equation of the ellipse having foci at, and, and a major ais of length, as shown in Figure.. Solution Because the foci occur at, and,, the center of the ellipse is, ) and the distance from the center to one of the foci is c. Because a, ou know that a. Now, from c a b, ou have b a c 5. Because the major ais is horizontal, the standard equation is. 5 This equation simplifies to 9 5. Now tr Eercise. Eample Sketching an Ellipse ( + ) ( ) + = ( 5, ) (, ) (, ) (, ) (, ) ( +, ) 5 (, ) FIGURE.5 Sketch the ellipse given b 8 9. Solution Begin b writing the original equation in standard form. In the fourth step, note that 9 and are added to both sides of the equation when completing the squares Write original equation. Group terms Factor out of -terms. Write in completed square form. Divide each side b. Write in standard form. From this standard form, it follows that the center is h, k,. Because the denominator of the -term is a, the endpoints of the major ais lie two units to the right and left of the center. Similarl, because the denominator of the -term is b, the endpoints of the minor ais lie one unit up and down from the center. Now, from c a b, ou have c. So, the foci of the ellipse are, and,. The ellipse is shown in Figure.5. Now tr Eercise 7.

21 Section. Ellipses 75 Eample Analzing an Ellipse ( ) ( + ) + = Verte, + Focus ( ( (, ), FIGURE. ( ( Center Focus Verte (, ) (, ) Find the center, vertices, and foci of the ellipse 8 8. Solution B completing the square, ou can write the original equation in standard form Write original equation. Group terms. The major ais is vertical, where h, k, a, b, and c a b. So, ou have the following. 8 8 Center:, Vertices:, Foci: The graph of the ellipse is shown in Figure.. Now tr Eercise 5., Factor out of -terms. Write in completed square form. Divide each side b. Write in standard form.,, TECHNOLOGY You can use a graphing utilit to graph an ellipse b graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Eample, first solve for to get and Use a viewing window in which 9 and 7. You should obtain the graph shown below.. 9 7

22 7 Chapter Topics in Analtic Geometr Application Ellipses have man practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Eample investigates the elliptical orbit of the moon about Earth. Eample An Application Involving an Elliptical Orbit 77, km Moon The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure.7. The major and minor aes of the orbit have lengths of 78,8 kilometers and 77, kilometers, respectivel. Find the greatest and smallest distances (the apogee and perigee, respectivel) from Earth s center to the moon s center. Earth Perigee FIGURE.7 78,8 km Apogee Solution Because a 78,8 and b 77,, ou have a 8, and b 8,8 which implies that c a b 8, 8,8,8. So, the greatest distance between the center of Earth and the center of the moon is WARNING / CAUTION Note in Eample and Figure.7 that Earth is not the center of the moon s orbit. a c 8,,8 5,58 kilometers and the smallest distance is a c 8,,8,9 kilometers. Now tr Eercise 5. Eccentricit One of the reasons it was difficult for earl astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetar orbits are relativel close to their centers, and so the orbits are nearl circular. To measure the ovalness of an ellipse, ou can use the concept of eccentricit. Definition of Eccentricit The eccentricit e of an ellipse is given b the ratio e c a. Note that < e < for ever ellipse.

23 Section. Ellipses 77 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major ais between the vertices and the center, it follows that < c < a. For an ellipse that is nearl circular, the foci are close to the center and the ratio c a is small, as shown in Figure.8. On the other hand, for an elongated ellipse, the foci are close to the vertices and the ratio c a is close to, as shown in Figure.9. Foci Foci c e= a c e= c a e is small. c e is close to. a FIGURE a.8 FIGURE.9 The orbit of the moon has an eccentricit of e ".59, and the eccentricities of the eight planetar orbits are as follows. Jupiter: Saturn: e ".8 e ".5 Earth: e ".7 Uranus: e ".7 Mars: e ".9 Neptune: e ".8 NASA Mercur: e ".5 Venus: e ".8 The time it takes Saturn to orbit the sun is about 9. Earth ears. CLASSROOM DISCUSSION Ellipses and Circles a. Show that the equation of an ellipse can be written as! h!! k! " a a! e!. b. For the equation in part (a), let a, h, and k, and use a graphing utilit to graph the ellipse for e.95, e.75, e.5, e.5, and e.. Discuss the changes in the shape of the ellipse as e approaches. c. Make a conjecture about the shape of the graph in part (b) when e. What is the equation of this ellipse? What is another name for an ellipse with an eccentricit of?

24 78 Chapter Topics in Analtic Geometr. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. An is the set of all points, in a plane, the sum of whose distances from two distinct fied points, called, is constant.. The chord joining the vertices of an ellipse is called the, and its midpoint is the of the ellipse.. The chord perpendicular to the major ais at the center of the ellipse is called the of the ellipse.. The concept of is used to measure the ovalness of an ellipse. SKILLS AND APPLICATIONS In Eercises 5, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) (b) (d) (f) In Eercises 8, find the standard form of the equation of the ellipse with the given characteristics and center at the origin Vertices: ±7, ; foci: ±,. Vertices:, ±8; foci:, ± 5. Foci: ±5, ; major ais of length. Foci: ±, ; major ais of length 7. Vertices:, ±5; passes through the point, 8. Vertical major ais; passes through the points, and, In Eercises 9 8, find the standard form of the equation of the ellipse with the given characteristics. 9.. (, ) (, ) 5 8 (, ) (, ) (, ) 8 (, ) (, ) (, ) 5 (, ) (, ). Vertices:,, 8, ; minor ais of length. Foci:,,, ; major ais of length (, ) (, ). Foci:,,, 8; major ais of length. Center:, ; verte:, ; minor ais of length 5. Center:, ; a c; vertices:,,,. Center:, ; a c; foci:,, 5, (, ) (, ) (, ) (, )

25 Section. Ellipses Vertices:,,, ; endpoints of the minor ais:,,, 8. Vertices: 5,, 5, ; endpoints of the minor ais:,, 9, In Eercises 9 5, identif the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricit of the conic (if applicable), and sketch its graph In Eercises 5 5, use a graphing utilit to graph the ellipse. Find the center, foci, and vertices. (Recall that it ma be necessar to solve the equation for and obtain two equations.) In Eercises 57, find the eccentricit of the ellipse Find an equation of the ellipse with vertices ±5, and eccentricit e 5.. Find an equation of the ellipse with vertices, ±8 and eccentricit e.. ARCHITECTURE A semielliptical arch over a tunnel for a one-wa road through a mountain has a major ais of 5 feet and a height at the center of feet. (a) Draw a rectangular coordinate sstem on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identif the coordinates of the known points. (b) Find an equation of the semielliptical arch. (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch?. ARCHITECTURE A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of feet at the center and a width of feet along the base (see figure). The contractor draws the outline of the ellipse using tacks as described at the beginning of this section. Determine the required positions of the tacks and the length of the string COMET ORBIT Halle s comet has an elliptical orbit, with the sun at one focus. The eccentricit of the orbit is approimatel.97. The length of the major ais of the orbit is approimatel 5.88 astronomical units. (An astronomical unit is about 9 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major ais on the -ais. (b) Use a graphing utilit to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun s center to the comet s center.

26 75 Chapter Topics in Analtic Geometr. SATELLITE ORBIT The first artificial satellite to orbit Earth was Sputnik I (launched b the former Soviet Union in 957). Its highest point above Earth s surface was 97 kilometers, and its lowest point was 8 kilometers (see figure). The center of Earth was at one focus of the elliptical orbit, and the radius of Earth is 78 kilometers. Find the eccentricit of the orbit. EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The graph of is an ellipse. 7. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricit of the ellipse is large (close to ). 75. Consider the ellipse 7. MOTION OF A PENDULUM The relation between the velocit (in radians per second) of a pendulum and its angular displacement from the vertical can be modeled b a semiellipse. A -centimeter pendulum crests when the angular displacement is. radian and. radian. When the pendulum is at equilibrium the velocit is. radians per second., Focus 8 km 97 km (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum? 8. GEOMETRY A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major ais is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it ields other points on the curve (see figure). Show that the length of each latus rectum is b a. F F Latera recta a, b a b. (a) The area of the ellipse is given b A ab. Write the area of the ellipse as a function of a. (b) Find the equation of an ellipse with an area of square centimeters. (c) Complete the table using our equation from part (a), and make a conjecture about the shape of the ellipse with maimum area. a 8 9 A (d) Use a graphing utilit to graph the area function and use the graph to support our conjecture in part (c). 7. THINK ABOUT IT At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks, a string of fied length (greater than the distance between the two tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced b the pencil is an ellipse. (a) What is the length of the string in terms of a? (b) Eplain wh the path is an ellipse. 77. THINK ABOUT IT Find the equation of an ellipse such that for an point on the ellipse, the sum of the distances from the point, and, is. 78. CAPSTONE Describe the relationship between circles and ellipses. How are the similar? How do the differ? In Eercises 9 7, sketch the graph of the ellipse, using latera recta (see Eercise 8) PROOF Show that a b c for the ellipse a b where a >, b >, and the distance from the center of the ellipse, to a focus is c.

27 Section. Hperbolas 75. HYPERBOLAS What ou should learn Write equations of hperbolas in standard form. Find asmptotes of and graph hperbolas. Use properties of hperbolas to solve real-life problems. Classif conics from their general equations. Wh ou should learn it Hperbolas can be used to model and solve man tpes of real-life problems. For instance, in Eercise 5 on page 759, hperbolas are used in long distance radio navigation for aircraft and ships. Introduction The third tpe of conic is called a hperbola. The definition of a hperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fied, whereas for a hperbola the difference of the distances between the foci and a point on the hperbola is fied. Definition of Hperbola A hperbola is the set of all points, in a plane, the difference of whose distances from two distinct fied points (foci) is a positive constant. See Figure.. Focus d (, ) d Focus d d is a positive constant. Branch FIGURE. FIGURE. Verte a Center c Transverse ais Verte Branch U.S. Nav, William Lipski/AP Photo The graph of a hperbola has two disconnected branches. The line through the two foci intersects the hperbola at its two vertices. The line segment connecting the vertices is the transverse ais, and the midpoint of the transverse ais is the center of the hperbola. See Figure.. The development of the standard form of the equation of a hperbola is similar to that of an ellipse. Note in the definition below that a, b, and c are related differentl for hperbolas than for ellipses. Standard Equation of a Hperbola The standard form of the equation of a hperbola with center h, k is h a k a Transverse ais is horizontal. Transverse ais is vertical. The vertices are a units from the center, and the foci are c units from the center. Moreover, c a b. If the center of the hperbola is at the origin,, the equation takes one of the following forms. a b k b h b. Transverse ais is horizontal. a b Transverse ais is vertical.

28 75 Chapter Topics in Analtic Geometr Figure. shows both the horizontal and vertical orientations for a hperbola. ( h) ( k) = a b ( k) ( h) = a b ( h, k + c) ( h c, k) ( h, k) ( h + c, k) ( h, k) Transverse ais is horizontal. FIGURE. ( h, k c) Transverse ais is vertical. Eample Finding the Standard Equation of a Hperbola When finding the standard form of the equation of an conic, it is helpful to sketch a graph of the conic with the given characteristics. Find the standard form of the equation of the hperbola with foci, and 5, and vertices, and,. Solution B the Midpoint Formula, the center of the hperbola occurs at the point,. Furthermore, c 5 and a, and it follows that b c a 9 5. So, the hperbola has a horizontal transverse ais and the standard form of the equation is 5. This equation simplifies to 5. See Figure.. ( ) ( ) = ( 5 ( 5 (, ) (, ) (, ) (, ) (5, ) FIGURE. Now tr Eercise 5.

29 Section. Hperbolas 75 Conjugate ais (h, k + b) Asmptote Asmptotes of a Hperbola Each hperbola has two asmptotes that intersect at the center of the hperbola, as shown in Figure.. The asmptotes pass through the vertices of a rectangle of dimensions a b b, with its center at h, k. The line segment of length b joining h, k b and h, k b or h b, k and h b, k is the conjugate ais of the hperbola. FIGURE. (h, k) (h, k b) (h a, k) (h + a, k) Asmptote Asmptotes of a Hperbola The equations of the asmptotes of a hperbola are k ± b h a k ± a h. b Transverse ais is horizontal. Transverse ais is vertical. Eample Using Asmptotes to Sketch a Hperbola Sketch the hperbola whose equation is. Algebraic Solution Divide each side of the original equation b, and rewrite the equation in standard form. Write in standard form. From this, ou can conclude that a, b, and the transverse ais is horizontal. So, the vertices occur at, and,, and the endpoints of the conjugate ais occur at, and,. Using these four points, ou are able to sketch the rectangle shown in Figure.5. Now, from c a b, ou have c 5. So, the foci of the hperbola are 5, and 5,. Finall, b drawing the asmptotes through the corners of this rectangle, ou can complete the sketch shown in Figure.. Note that the asmptotes are and. Graphical Solution Solve the equation of the hperbola for as follows. ± Then use a graphing utilit to graph and in the same viewing window. Be sure to use a square setting. From the graph in Figure.7, ou can see that the transverse ais is horizontal. You can use the zoom and trace features to approimate the vertices to be, and,. = 8 (, ) = (, ) (, ) (, ) ( 5, ) ( 5, ) = FIGURE.7 FIGURE.5 FIGURE. Now tr Eercise.

30 75 Chapter Topics in Analtic Geometr Eample Finding the Asmptotes of a Hperbola Sketch the hperbola given b 8 and find the equations of its asmptotes and the foci. Solution 8 8 Write original equation. Group terms. Factor from -terms. Add to each side. Write in completed square form. (, 7) 5 (, ) ( + ) = (, ) ( ) 5 (, ) (, 7 ) FIGURE.8 Divide each side b. Write in standard form. From this equation ou can conclude that the hperbola has a vertical transverse ais, centered at,, has vertices, and,, and has a conjugate ais with endpoints, and,. To sketch the hperbola, draw a rectangle through these four points. The asmptotes are the lines passing through the corners of the rectangle. Using a and b, ou can conclude that the equations of the asmptotes are and Finall, ou can determine the foci b using the equation c a b. So, ou have c 7, and the foci are,7 and, 7. The hperbola is shown in Figure.8. Now tr Eercise 9.. TECHNOLOGY You can use a graphing utilit to graph a hperbola b graphing the upper and lower portions in the same viewing window. For instance, to graph the hperbola in Eample, first solve for to get and. Use a viewing window in which 9 9 and. You should obtain the graph shown below. Notice that the graphing utilit does not draw the asmptotes. However, if ou trace along the branches, ou will see that the values of the hperbola approach the asmptotes. 9 9

31 Section. Hperbolas 755 = 8 Eample Using Asmptotes to Find the Standard Equation (, ) (, 5) Find the standard form of the equation of the hperbola having vertices, 5 and, and having asmptotes 8 and as shown in Figure.9. Solution B the Midpoint Formula, the center of the hperbola is,. Furthermore, the hperbola has a vertical transverse ais with a. From the original equations, ou can determine the slopes of the asmptotes to be FIGURE.9 = + m and m a a b b and, because a, ou can conclude a b b b. So, the standard form of the equation is. Now tr Eercise. As with ellipses, the eccentricit of a hperbola is e c a Eccentricit and because c > a, it follows that e >. If the eccentricit is large, the branches of the hperbola are nearl flat, as shown in Figure.. If the eccentricit is close to, the branches of the hperbola are more narrow, as shown in Figure.. e is large. e is close to. Verte Focus Verte Focus e = c a c e = c a a c a FIGURE. FIGURE.

32 75 Chapter Topics in Analtic Geometr Applications The following application was developed during World War II. It shows how the properties of hperbolas can be used in radar and other detection sstems. Eample 5 An Application Involving Hperbolas Two microphones, mile apart, record an eplosion. Microphone A receives the sound seconds before microphone B. Where did the eplosion occur? (Assume sound travels at feet per second.) Solution Assuming sound travels at feet per second, ou know that the eplosion took place feet farther from B than from A, as shown in Figure.. The locus of all points that are feet closer to A than to B is one branch of the hperbola B A where a b c 58 c a c a c = 58 + ( c a) = 58 FIGURE. and a. So, b c a 5,759,, and ou can conclude that the eplosion occurred somewhere on the right branch of the hperbola,,. 5,759, Now tr Eercise 5. Hperbolic orbit Verte Elliptical orbit Sun p Parabolic orbit FIGURE. Another interesting application of conic sections involves the orbits of comets in our solar sstem. Of the comets identified prior to 97, 5 have elliptical orbits, 95 have parabolic orbits, and 7 have hperbolic orbits. The center of the sun is a focus of each of these orbits, and each orbit has a verte at the point where the comet is closest to the sun, as shown in Figure.. Undoubtedl, there have been man comets with parabolic or hperbolic orbits that were not identified. We onl get to see such comets once. Comets with elliptical orbits, such as Halle s comet, are the onl ones that remain in our solar sstem. If p is the distance between the verte and the focus (in meters), and v is the velocit of the comet at the verte (in meters per second), then the tpe of orbit is determined as follows.. Ellipse:. Parabola: v < GMp v GMp. Hperbola: v > GMp In each of these relations, M.989 kilograms (the mass of the sun) and G.7 cubic meter per kilogram-second squared (the universal gravitational constant).

33 Section. Hperbolas 757 General Equations of Conics Classifing a Conic from Its General Equation The graph of A C D E F is one of the following.. Circle: A C. Parabola: AC A or C, but not both.. Ellipse: AC > A and C have like signs.. Hperbola: AC < A and C have unlike signs. The test above is valid if the graph is a conic. The test does not appl to equations such as, whose graph is not a conic. Eample Classifing Conics from General Equations Classif the graph of each equation. a. 9 5 b. 8 c. d. 8 Solution a. For the equation 9 5, ou have AC. Parabola So, the graph is a parabola. b. For the equation 8, ou have HISTORICAL NOTE AC <. Hperbola So, the graph is a hperbola. c. For the equation, ou have The Granger Collection AC >. Caroline Herschel (75 88) was the first woman to be credited with detecting a new comet. During her long life, this English astronomer discovered a total of eight new comets. Ellipse So, the graph is an ellipse. d. For the equation 8, ou have A C. Circle So, the graph is a circle. Now tr Eercise. CLASSROOM DISCUSSION Sketching Conics Sketch each of the conics described in Eample. Write a paragraph describing the procedures that allow ou to sketch the conics efficientl.

34 758 Chapter Topics in Analtic Geometr. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. A is the set of all points, in a plane, the difference of whose distances from two distinct fied points, called, is a positive constant.. The graph of a hperbola has two disconnected parts called.. The line segment connecting the vertices of a hperbola is called the, and the midpoint of the line segment is the of the hperbola.. Each hperbola has two that intersect at the center of the hperbola. SKILLS AND APPLICATIONS In Eercises 5 8, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d) In Eercises 9, find the center, vertices, foci, and the equations of the asmptotes of the hperbola, and sketch its graph using the asmptotes as an aid In Eercises 8, find the center, vertices, foci, and the equations of the asmptotes of the hperbola. Use a graphing utilit to graph the hperbola and its asmptotes In Eercises 9, find the standard form of the equation of the hperbola with the given characteristics and center at the origin. 9. Vertices:, ±; foci:, ±. Vertices: ±, ; foci: ±,. Vertices: ±, ; asmptotes: ±5. Vertices:, ±; asmptotes: ±. Foci:, ±8; asmptotes: ±. Foci: ±, ; asmptotes: ± In Eercises 5, find the standard form of the equation of the hperbola with the given characteristics. 5. Vertices:,,, ; foci:,, 8,. Vertices:,,, ; foci:,,, 7. Vertices:,,, 9; foci:,,, 8. Vertices:,,, ); foci:,,,

35 Section. Hperbolas Vertices:,,, ; passes through the point, 5. Vertices:,,, ; passes through the point 5,. Vertices:,,, ; passes through the point 5,. Vertices:,,, ; passes through the point,5. Vertices:,,, ; asmptotes:,. Vertices:,,, ; asmptotes:, 5. Vertices:,,, ; asmptotes:,. Vertices:,,, ; asmptotes:, In Eercises 7 5, write the standard form of the equation of the hperbola (, 5) (, ) 8 8 (, ) (, ) (, ) (, ) (5, ) 5. ART A sculpture has a hperbolic cross section (see figure). (, ) (, ) 8 8 (, ) 8 (, ) 8 ( 8, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (a) Write an equation that models the curved sides of the sculpture. (b) Each unit in the coordinate plane represents foot. Find the width of the sculpture at a height of 5 feet. 5. SOUND LOCATION You and a friend live miles apart (on the same east-west street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 8 seconds later our friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate sstem is measured in feet and that sound travels at feet per second.) 5. SOUND LOCATION Three listening stations located at,,,, and, monitor an eplosion. The last two stations detect the eplosion second and seconds after the first, respectivel. Determine the coordinates of the eplosion. (Assume that the coordinate sstem is measured in feet and that sound travels at feet per second.) 5. LORAN Long distance radio navigation for aircraft and ships uses snchronized pulses transmitted b widel separated transmitting stations. These pulses travel at the speed of light (8, miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hperbola having the transmitting stations as foci. Assume that two stations, miles apart, are positioned on the rectangular coordinate sstem at points with coordinates 5, and 5,, and that a ship is traveling on a hperbolic path with coordinates, 75 (see figure). 5 Station Station Not drawn to scale Ba (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is microseconds (. second). (b) Determine the distance between the ship and station when the ship reaches the shore. (c) The ship wants to enter a ba located between the two stations. The ba is miles from station. What should be the time difference between the pulses? (d) The ship is miles offshore when the time difference in part (c) is obtained. What is the position of the ship?

36 7 Chapter Topics in Analtic Geometr 55. PENDULUM The base for a pendulum of a clock has the shape of a hperbola (see figure). (a) Write an equation of the cross section of the base. (b) Each unit in the coordinate plane represents foot. Find the width of the base of the pendulum inches from the bottom. 5. HYPERBOLIC MIRROR A hperbolic mirror (used in some telescopes) has the propert that a light ra directed at a focus will be reflected to the other focus. The focus of a hperbolic mirror (see figure) has coordinates,. Find the verte of the mirror if the mount at the top edge of the mirror has coordinates,. In Eercises 57 7, classif the graph of the equation as a circle, a parabola, an ellipse, or a hperbola (, 9) (, ) (, 9) (, ) (, 9) (, ) (, 9) (, ) (, ) EXPLORATION TRUE OR FALSE? In Eercises 7 7, determine whether the statement is true or false. Justif our answer. 7. In the standard form of the equation of a hperbola, the larger the ratio of b to a, the larger the eccentricit of the hperbola. 7. In the standard form of the equation of a hperbola, the trivial solution of two intersecting lines occurs when b. 75. If D and E, then the graph of D E is a hperbola. 7. If the asmptotes of the hperbola, where a b a, b >, intersect at right angles, then a b. 77. Consider a hperbola centered at the origin with a horizontal transverse ais. Use the definition of a hperbola to derive its standard form. 78. WRITING Eplain how the central rectangle of a hperbola can be used to sketch its asmptotes. 79. THINK ABOUT IT Change the equation of the hperbola so that its graph is the bottom half of the hperbola CAPSTONE Given the hperbolas 9 and 9 8. A circle and a parabola can have,,,, or points of intersection. Sketch the circle given b. Discuss how this circle could intersect a parabola with an equation of the form C. Then find the values of C for each of the five cases described below. Use a graphing utilit to verif our results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection describe an common characteristics that the hperbolas share, as well as an differences in the graphs of the hperbolas. Verif our results b using a graphing utilit to graph each of the hperbolas in the same viewing window.

37 Section.5 Rotation of Conics 7.5 ROTATION OF CONICS What ou should learn Rotate the coordinate aes to eliminate the -term in equations of conics. Use the discriminant to classif conics. Wh ou should learn it As illustrated in Eercises on page 77, rotation of the coordinate aes can help ou identif the graph of a general second-degree equation. Rotation In the preceding section, ou learned that the equation of a conic with aes parallel to one of the coordinate aes has a standard form that can be written in the general form A C D E F. Horizontal or vertical ais In this section, ou will stud the equations of conics whose aes are rotated so that the are not parallel to either the -ais or the -ais. The general equation for such conics contains an -term. A B C D E F Equation in -plane To eliminate this -term, ou can use a procedure called rotation of aes. The objective is to rotate the - and -aes until the are parallel to the aes of the conic. The rotated aes are denoted as the -ais and the -ais, as shown in Figure.. θ FIGURE. After the rotation, the equation of the conic in the new A C D E F. -plane will have the form Equation in -plane Because this equation has no -term, ou can obtain a standard form b completing the square. The following theorem identifies how much to rotate the aes to eliminate the -term and also the equations for determining the new coefficients A, C, D, E, and F. Rotation of Aes to Eliminate an -Term The general second-degree equation A B C D E F can be rewritten as A C D E F b rotating the coordinate aes through an angle, where cot A C B. The coefficients of the new equation are obtained b making the substitutions cos sin and sin cos.

38 7 Chapter Topics in Analtic Geometr WARNING / CAUTION Remember that the substitutions and were developed to eliminate the -term in the rotated sstem. You can use this as a check on our work. In other words, if our final equation contains an -term, ou know that ou have made a mistake. cos sin sin cos Eample Rotation of Aes for a Hperbola Write the equation in standard form. Solution Because A, B, and C, ou have which implies that and cot A C B cos sin sin cos. ( ) ( ) = ( ( = Vertices: In -sstem:,,, In -sstem:,,, FIGURE.5 ( ( The equation in the equation. -sstem is obtained b substituting these epressions in the Write in standard form. In the -sstem, this is a hperbola centered at the origin with vertices at ±,, as shown in Figure.5. To find the coordinates of the vertices in the -sstem, substitute the coordinates ±, in the equations and This substitution ields the vertices, and, in the -sstem. Note also that the asmptotes of the hperbola have equations which correspond to the original - and -aes. Now tr Eercise.. ±,

39 Section.5 Rotation of Conics 7 Eample Rotation of Aes for an Ellipse Sketch the graph of = Vertices: In -sstem: ±, In -sstem:,,, FIGURE. ( ) ( ) + = Solution Because A 7, B, and C, ou have which implies that The equation in the -sstem is obtained b making the substitutions and in the original equation. So, ou have which simplifies to Write in standard form. This is the equation of an ellipse centered at the origin with vertices ±, in the -sstem, as shown in Figure.. cot A C B cos sin sin cos Now tr Eercise 9.

40 7 Chapter Topics in Analtic Geometr Eample Rotation of Aes for a Parabola 5 θ FIGURE.7 Sketch the graph of 55. Solution Because A, B, and C, ou have Using this information, draw a right triangle as shown in Figure.7. From the figure, ou can see that cos 5. To find the values of sin and cos, ou can use the half-angle formulas in the forms So, cot A C B sin cos and sin cos cos cos Consequentl, ou use the substitutions cos sin cos cos = ( + ) = ( )( 5) Verte: In -sstem: 5, In -sstem: FIGURE.8 55, 55 θ. sin cos Substituting these epressions in the original equation, ou have which simplifies as follows. 5 5 Group terms. Write in completed square form. Write in standard form. The graph of this equation is a parabola with verte 5,. Its ais is parallel to the -ais in the -sstem, and because sin 5, as shown in Figure Now tr Eercise ,

41 Section.5 Rotation of Conics 75 Invariants Under Rotation In the rotation of aes theorem listed at the beginning of this section, note that the constant term is the same in both equations, Such quantities are invariant under rotation. The net theorem lists some other rotation invariants. Rotation Invariants F F. The rotation of the coordinate aes through an angle that transforms the equation A B C D E F into the form A C D E F has the following rotation invariants.. F F. A C A C. B AC B AC WARNING / CAUTION If there is an -term in the equation of a conic, ou should realize then that the conic is rotated. Before rotating the aes, ou should use the discriminant to classif the conic. You can use the results of this theorem to classif the graph of a second-degree equation with an -term in much the same wa ou do for a second-degree equation without an -term. Note that because the invariant B AC reduces to B AC AC. Discriminant This quantit is called the discriminant of the equation A B C D E F. Now, from the classification procedure given in Section., ou know that the sign of determines the tpe of graph for the equation AC B, A C D E F. Consequentl, the sign of B AC will determine the tpe of graph for the original equation, as given in the following classification. Classification of Conics b the Discriminant The graph of the equation A B C D E F is, ecept in degenerate cases, determined b its discriminant as follows.. Ellipse or circle: B AC <. Parabola: B AC. Hperbola: B AC > For eample, in the general equation ou have A, B 7, and C 5. So the discriminant is B AC Because <, the graph of the equation is an ellipse or a circle.

42 7 Chapter Topics in Analtic Geometr Eample Rotation and Graphing Utilities For each equation, classif the graph of the equation, use the Quadratic Formula to solve for, and then use a graphing utilit to graph the equation. a. b. 9 c. 8 7 Solution a. Because B AC 9 <, the graph is a circle or an ellipse. Solve for as follows. Write original equation. ± ± 7 Graph both of the equations to obtain the ellipse shown in Figure Top half of ellipse 7 Bottom half of ellipse b. Because B AC, the graph is a parabola. FIGURE Quadratic form a b c Write original equation. Quadratic form a b c ± 9 9 Graphing both of the equations to obtain the parabola shown in Figure.5. c. Because B AC 8 >, the graph is a hperbola. 8 7 FIGURE ± Write original equation. Quadratic form a b c 8 7 The graphs of these two equations ield the hperbola shown in Figure.5. Now tr Eercise. 5 5 CLASSROOM DISCUSSION FIGURE.5 Classifing a Graph as a Hperbola In Section., it was mentioned that the graph of f!" / is a hperbola. Use the techniques in this section to verif this, and justif each step. Compare our results with those of another student.

43 Section.5 Rotation of Conics 77.5 EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. After rotating the coordinate aes through an angle, the general second-degree equation in the new -plane will have the form.. The procedure used to eliminate the -term in a general second-degree equation is called of.. Quantities that are equal in both the original equation of a conic and the equation of the rotated conic are.. The quantit B AC is called the of the equation A B C D E F. SKILLS AND APPLICATIONS In Eercises 5, the -coordinate sstem has been rotated degrees from the -coordinate sstem. The coordinates of a point in the -coordinate sstem are given. Find the coordinates of the point in the rotated coordinate sstem In Eercises, rotate the aes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of aes In Eercises 7, use a graphing utilit to graph the conic. Determine the angle through which the aes are rotated. Eplain how ou used the graphing utilit to obtain the graph ,,,, 5,,,, ,,,, 5,,,, In Eercises 7, match the graph with its equation. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) (b) (d) (f)

44 78 Chapter Topics in Analtic Geometr In Eercises 5, (a) use the discriminant to classif the graph, (b) use the Quadratic Formula to solve for, and (c) use a graphing utilit to graph the equation In Eercises 5 5, sketch (if possible) the graph of the degenerate conic In Eercises 57 7, find an points of intersection of the graphs algebraicall and then verif using a graphing utilit EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. The graph of the equation k where k is an constant less than is a hperbola. 7. After a rotation of aes is used to eliminate the -term from an equation of the form A B C D E F the coefficients of the - and -terms remain A and C, respectivel. 7. Show that the equation r is invariant under rotation of aes. 75. Find the lengths of the major and minor aes of the ellipse graphed in Eercise., 7. CAPSTONE Consider the equation 5. (a) Without calculating, eplain how to rewrite the equation so that it does not have an -term. (b) Eplain how to identif the graph of the equation.

45 Section. Parametric Equations 79. PARAMETRIC EQUATIONS Jed Jacobsohn/Gett Images What ou should learn Evaluate sets of parametric equations for given values of the parameter. Sketch curves that are represented b sets of parametric equations. Rewrite sets of parametric equations as single rectangular equations b eliminating the parameter. Find sets of parametric equations for graphs. Wh ou should learn it Parametric equations are useful for modeling the path of an object. For instance, in Eercise on page 775, ou will use a set of parametric equations to model the path of a baseball. Plane Curves Up to this point ou have been representing a graph b a single equation involving the two variables and. In this section, ou will stud situations in which it is useful to introduce a third variable to represent a curve in the plane. To see the usefulness of this procedure, consider the path followed b an object that is propelled into the air at an angle of 5. If the initial velocit of the object is 8 feet per second, it can be shown that the object follows the parabolic path 7 Rectangular equation as shown in Figure.5. However, this equation does not tell the whole stor. Although it does tell ou where the object has been, it does not tell ou when the object was at a given point, on the path. To determine this time, ou can introduce a third variable t, called a parameter. It is possible to write both and as functions of t to obtain the parametric equations t t t. Parametric equation for Parametric equation for From this set of equations ou can determine that at time t, the object is at the point,. Similarl, at time t, the object is at the point,, and so on, as shown in Figure.5. Rectangular equation: = + t = 8 7 (, 8) Parametric equations: = t = t + t 9 t = (, ) t = (7, ) Curvilinear Motion: Two Variables for Position, One Variable for Time FIGURE.5 For this particular motion problem, and are continuous functions of t, and the resulting path is a plane curve. (Recall that a continuous function is one whose graph can be traced without lifting the pencil from the paper.) Definition of Plane Curve If f and g are continuous functions of t on an interval I, the set of ordered pairs ft, gt is a plane curve C. The equations ft and gt are parametric equations for C, and t is the parameter.

46 77 Chapter Topics in Analtic Geometr Sketching a Plane Curve When sketching a curve represented b a pair of parametric equations, ou still plot points in the -plane. Each set of coordinates, is determined from a value chosen for the parameter t. Plotting the resulting points in the order of increasing values of t traces the curve in a specific direction. This is called the orientation of the curve. Eample Sketching a Curve WARNING / CAUTION When using a value of t to find, be sure to use the same value of t to find the corresponding value of. Organizing our results in a table, as shown in Eample, can be helpful. Sketch the curve given b the parametric equations t and t, t. Solution Using values of t in the specified interval, the parametric equations ield the points, shown in the table. t t = t = t = FIGURE.5 t = t = t = FIGURE.5 t = = t = t t = t = t = = t = t t t = t = t B plotting these points in the order of increasing t, ou obtain the curve C shown in Figure.5. Note that the arrows on the curve indicate its orientation as t increases from to. So, if a particle were moving on this curve, it would start at, and then move along the curve to the point 5,. Now tr Eercises 5(a) and (b). Note that the graph shown in Figure.5 does not define as a function of. This points out one benefit of parametric equations the can be used to represent graphs that are more general than graphs of functions. It often happens that two different sets of parametric equations have the same graph. For eample, the set of parametric equations t and t, t has the same graph as the set given in Eample. However, b comparing the values of t in Figures.5 and.5, ou can see that this second graph is traced out more rapidl (considering t as time) than the first graph. So, in applications, different parametric representations can be used to represent various speeds at which objects travel along a given path. 5

47 Section. Parametric Equations 77 Eliminating the Parameter Eample uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified b finding a rectangular equation (in and ) that has the same graph. This process is called eliminating the parameter. Parametric equations Solve for t in one equation. Substitute in other equation. Rectangular equation t t t Now ou can recognize that the equation represents a parabola with a horizontal ais and verte at,. When converting equations from parametric to rectangular form, ou ma need to alter the domain of the rectangular equation so that its graph matches the graph of the parametric equations. Such a situation is demonstrated in Eample. Eample Eliminating the Parameter Sketch the curve represented b the equations t and t t b eliminating the parameter and adjusting the domain of the resulting rectangular equation. Parametric equations: =, = t t + t + FIGURE.55 t = t = t =.75 Solution Solving for t in the equation for produces which implies that t Now, substituting in the equation for, ou obtain the rectangular equation t. t t t From this rectangular equation, ou can recognize that the curve is a parabola that opens downward and has its verte at,. Also, this rectangular equation is defined for all values of, but from the parametric equation for ou can see that the curve is defined onl when t >. This implies that ou should restrict the domain of to positive values, as shown in Figure.55. Now tr Eercise 5(c)..

48 77 Chapter Topics in Analtic Geometr To eliminate the parameter in equations involving trigonometric functions, tr using identities such as or sin sec cos tan as shown in Eample. It is not necessar for the parameter in a set of parametric equations to represent time. The net eample uses an angle as the parameter. Eample Sketch the curve represented b cos Eliminating an Angle Parameter and b eliminating the parameter. Solution sin, Begin b solving for cos and sin in the equations. θ = θ = FIGURE.5 θ = θ = = cos θ = sin θ cos Use the identit sin cos sin 9 and sin Solve for cos and cos to form an equation involving onl and. Pthagorean identit Substitute for cos and for sin. Rectangular equation From this rectangular equation, ou can see that the graph is an ellipse centered at,, with vertices, and, and minor ais of length b, as shown in Figure.5. Note that the elliptic curve is traced out counterclockwise as varies from to. Now tr Eercise 7. sin. In Eamples and, it is important to realize that eliminating the parameter is primaril an aid to curve sketching. If the parametric equations represent the path of a moving object, the graph alone is not sufficient to describe the object s motion. You still need the parametric equations to tell ou the position, direction, and speed at a given time. Finding Parametric Equations for a Graph You have been studing techniques for sketching the graph represented b a set of parametric equations. Now consider the reverse problem that is, how can ou find a set of parametric equations for a given graph or a given phsical description? From the discussion following Eample, ou know that such a representation is not unique. That is, the equations t and t, t produced the same graph as the equations t and t, t. This is further demonstrated in Eample.

49 Section. Parametric Equations 77 = t = t t t = t = t = Eample Finding Parametric Equations for a Graph Find a set of parametric equations to represent the graph of, using the following parameters. a. t b. t Solution t = t = a. Letting t, ou obtain the parametric equations t and t. b. Letting t, ou obtain the parametric equations FIGURE.57 t and t t t. In Figure.57, note how the resulting curve is oriented b the increasing values of t. For part (a), the curve would have the opposite orientation. Now tr Eercise 5. Eample 5 Parametric Equations for a Ccloid Describe the ccloid traced out b a point P on the circumference of a circle of radius a as the circle rolls along a straight line in a plane. Solution As the parameter, let be the measure of the circle s rotation, and let the point P, begin at the origin. When P is at the origin; when P is at a maimum point a, a; and when P is back on the -ais at a,. From Figure.58, ou can see that APC 8. So, ou have,, sin sin8 sinapc AC a BD a, In Eample 5, PD represents the arc of the circle between points P and D. cos cos8 which implies that BD a sin and Because the circle rolls along the -ais, ou know that OD PD AP a cos. a. Furthermore, because BA DC a, ou have OD BD a a sin So, the parametric equations are cosapc AP a and a sin BA AP a a cos. and a cos. TECHNOLOGY a P = (, ) ( a, a) Ccloid: = a( θ sin θ), = a( cos θ) ( a, a) You can use a graphing utilit in parametric mode to obtain a graph similar to Figure.58 b graphing the following equations. X T T sin T Y T cos T a O A FIGURE.58 B θ C D a Now tr Eercise 7. (a, ) a (a, )

50 77 Chapter Topics in Analtic Geometr. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. If f and g are continuous functions of t on an interval I, the set of ordered pairs f t, gt is a C.. The of a curve is the direction in which the curve is traced out for increasing values of the parameter.. The process of converting a set of parametric equations to a corresponding rectangular equation is called the.. A curve traced b a point on the circumference of a circle as the circle rolls along a straight line in a plane is called a. SKILLS AND APPLICATIONS 5. Consider the parametric equations t and t. (a) Create a table of - and -values using t,,,, and. (b) Plot the points, generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation b eliminating the parameter. Sketch its graph. How do the graphs differ?. Consider the parametric equations cos and sin. (a) Create a table of - and -values using,,, and. (b) Plot the points, generated in part (a), and sketch a graph of the parametric equations. (c) Find the rectangular equation b eliminating the parameter. Sketch its graph. How do the graphs differ? In Eercises 7, (a) sketch the curve represented b the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessar. 7. t 8. t t t t t t 5. t. t sin t 9. t. t t. t. t t. t. t t t t t 7. cos 8. cos sin, 9. sin. cos cos. cos. 5 cos sin sin. e t. e t e t 5. t. ln t ln t In Eercises 7 and 8, determine how the plane curves differ from each other. 7. (a) t (b) cos t cos (c) e t (d) e t 8. (a) t (b) t (c) In Eercises 9, eliminate the parameter and obtain the standard form of the rectangular equation. (d) 9. Line through, and, : t, t. Circle:. Ellipse: e t t sin t. Hperbola: sin t h r cos, h a cos, h a sec, In Eercises, use the results of Eercises 9 to find a set of parametric equations for the line or conic.. Line: passes through, and,. Line: passes through, and, 5. Circle: center:, ; radius:. Circle: center: 5, ; radius: sin e t t e t t e t e t k r sin k b sin k b tan

51 Section. Parametric Equations Ellipse: vertices: ±5, ; foci: ±, 8. Ellipse: vertices:, 7,, ; foci: (, 5,, 9. Hperbola: vertices: ±, ; foci: ±5,. Hperbola: vertices: ±, ; foci: ±, In Eercises 8, find a set of parametric equations for the rectangular equation using (a) t and (b) t In Eercises 9 5, use a graphing utilit to graph the curve represented b the parametric equations. 9. Ccloid: 5. Ccloid: 5. Prolate ccloid: 5. Prolate ccloid: 5. Hpoccloid: 5. Curtate ccloid: 55. Witch of Agnesi: 5. Folium of Descartes: In Eercises 57, match the parametric equations with the correct graph and describe the domain and range. [The graphs are labeled (a), (b), (c), and (d).] (a) sin, sin, sin, sin, cos, 8 sin, cot, (b) cos cos sin t t, cos cos 8 cos sin t t 57. Lissajous curve: 58. Evolute of ellipse: 59. Involute of circle:. Serpentine curve: PROJECTILE MOTION A projectile is launched at a height of h feet above the ground at an angle of with the horizontal. The initial velocit is v feet per second, and the path of the projectile is modeled b the parametric equations v cos t and In Eercises and, use a graphing utilit to graph the paths of a projectile launched from ground level at each value of and v. For each case, use the graph to approimate the maimum height and the range of the projectile.. (a) v 88 feet per second (b) (c) (d) v feet per second v 88 feet per second v feet per second. (a) v 5 feet per second (b) (c) (d),, 5, 5, 5, 5,,, v feet per second v 5 feet per second v feet per second. SPORTS The center field fence in Yankee Stadium is 7 feet high and 8 feet from home plate. A baseball is hit at a point feet above the ground. It leaves the bat at an angle of degrees with the horizontal at a speed of miles per hour (see figure). ft cos, sin cot, sin cos θ cos, cos sin h v sin t t. 7 ft 8 ft sin sin cos Not drawn to scale (c) (d) (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utilit to graph the path of the baseball when Is the hit a home run? 5. (c) Use the graphing utilit to graph the path of the baseball when Is the hit a home run?. (d) Find the minimum angle required for the hit to be a home run.

52 77 Chapter Topics in Analtic Geometr. SPORTS An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an angle of 5 with the horizontal and at an initial speed of 5 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utilit to graph the path of the arrow and approimate its maimum height. (d) Find the total time the arrow is in the air. 5. PROJECTILE MOTION Eliminate the parameter from the parametric equations v cos t and for the motion of a projectile to show that the rectangular equation is. PATH OF A PROJECTILE The path of a projectile is given b the rectangular equation 7.. (a) Use the result of Eercise 5 to find h, v, and. Find the parametric equations of the path. (b) Use a graphing utilit to graph the rectangular equation for the path of the projectile. Confirm our answer in part (a) b sketching the curve represented b the parametric equations. (c) Use the graphing utilit to approimate the maimum height of the projectile and its range. 7. CURTATE CYCLOID A wheel of radius a units rolls along a straight line without slipping. The curve traced b a point P that is b units from the center b < a is called a curtate ccloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. a P θ b (, a b) a ( a, a + b) a h v sin t t sec tan h. v a t 8. EPICYCLOID A circle of radius one unit rolls around the outside of a circle of radius two units without slipping. The curve traced b a point on the circumference of the smaller circle is called an epiccloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. EXPLORATION θ (, ) TRUE OR FALSE? In Eercises 9 and 7, determine whether the statement is true or false. Justif our answer. 9. The two sets of parametric equations t, t and t, 9t have the same rectangular equation. 7. If is a function of t, and is a function of t, then must be a function of. 7. WRITING Write a short paragraph eplaining wh parametric equations are useful. 7. WRITING Eplain the process of sketching a plane curve given b parametric equations. What is meant b the orientation of the curve? 7. Use a graphing utilit set in parametric mode to enter the parametric equations from Eample. Over what values should ou let t var to obtain the graph shown in Figure.55? 7. CAPSTONE Consider the parametric equations 8 cos t and 8 sin t. (a) Describe the curve represented b the parametric equations. (b) How does the curve represented b the parametric equations 8 cos t and 8 sin t compare with the curve described in part (a)? (c) How does the original curve change when cosine and sine are interchanged?

53 Section.7 Polar Coordinates POLAR COORDINATES What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa. Wh ou should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Eercises 5 8 on page 78, ou are asked to find multiple representations of polar coordinates. Introduction So far, ou have been representing graphs of equations as collections of points, on the rectangular coordinate sstem, where and represent the directed distances from the coordinate aes to the point,. In this section, ou will stud a different sstem called the polar coordinate sstem. To form the polar coordinate sstem in the plane, fi a point O, called the pole (or origin), and construct from O an initial ra called the polar ais, as shown in Figure.59. Then each point P in the plane can be assigned polar coordinates r, as follows.. r directed distance from O to P. directed angle, counterclockwise from polar ais to segment OP O r = directed distance θ FIGURE.59 P = ( r, θ) = directed angle Polar ais Eample Plotting Points on the Polar Coordinate Sstem a. The point r,, lies two units from the pole on the terminal side of the angle as shown in Figure.. b. The point r,, lies three units from the pole on the terminal side of the angle as shown in Figure.. c. The point r,, coincides with the point,, as shown in Figure..,, θ =, ( ) FIGURE. θ = FIGURE. Now tr Eercise 7. ( ), FIGURE. ( ), θ =

54 778 Chapter Topics in Analtic Geometr In rectangular coordinates, each point, has a unique representation. This is not true for polar coordinates. For instance, the coordinates r, and r, represent the same point, as illustrated in Eample. Another wa to obtain multiple representations of a point is to use negative values for r. Because r is a directed distance, the coordinates r, and r, represent the same point. In general, the point r, can be represented as r, r, ± n or r, r, where n is an integer. Moreover, the pole is represented b,, where is an angle. ± n Eample Multiple Representations of Points ( ), θ = ( ) ( ) ( ) ( ), =, 5 =, 7 =, =... FIGURE. Plot the point, and find three additional polar representations of this point, using < <. Solution The point is shown in Figure.. Three other representations are as follows., 5,, 7,,, Now tr Eercise. Add to. Replace r b r; subtract Replace r b r; add to. from. Coordinate Conversion (r, θ) (, ) To establish the relationship between polar and rectangular coordinates, let the polar ais coincide with the positive -ais and the pole with the origin, as shown in Figure.. Because, lies on a circle of radius r, it follows that r. Moreover, for r >, the definitions of the trigonometric functions impl that r tan, cos r, and sin r. Pole θ (Origin) FIGURE. Polar ais (-ais) If r <, ou can show that the same relationships hold. Coordinate Conversion The polar coordinates r, are related to the rectangular coordinates, as follows. Polar-to-Rectangular Rectangular-to-Polar r cos tan r sin r

55 Section.7 Polar Coordinates 779 ( r, θ) = (, ) (, ) = (, ) FIGURE.5 (, ) = (, ) (r, θ) =, ( ) FIGURE. (, ) = (, ) FIGURE.7 ( ) ( r, θ) =, (, ) =, ( ) (r, θ) =, ( ) Eample Polar-to-Rectangular Conversion Convert each point to rectangular coordinates. a., b., Solution a. For the point r,,, ou have the following. The rectangular coordinates are,,. (See Figure.5.) b. For the point r,, ou have the following. The rectangular coordinates are Eample Now tr Eercise. Rectangular-to-Polar Conversion Convert each point to polar coordinates. a., b., Solution r cos cos r sin sin cos sin a. For the second-quadrant point,,, ou have tan Because lies in the same quadrant as,, use positive r. r So, one set of polar coordinates is r,,, as shown in Figure.. b. Because the point,, lies on the positive -ais, choose. and, r. This implies that one set of polar coordinates is Figure.7. Now tr Eercise.,,. r,,, as shown in

56 78 Chapter Topics in Analtic Geometr FIGURE.8 FIGURE.9 Equation Conversion B comparing Eamples and, ou can see that point conversion from the polar to the rectangular sstem is straightforward, whereas point conversion from the rectangular to the polar sstem is more involved. For equations, the opposite is true. To convert a rectangular equation to polar form, ou simpl replace b r cos and b r sin. For instance, the rectangular equation can be written in polar form as follows. Rectangular equation Polar equation Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuit. Eample 5 demonstrates several polar-to-rectangular conversions that enable ou to sketch the graphs of some polar equations. Eample 5 Converting Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r b. c. r sec Solution r sin r cos r sec a. The graph of the polar equation r consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of, as shown in Figure.8. You can confirm this b converting to rectangular form, using the relationship r. r Polar equation Rectangular equation b. The graph of the polar equation consists of all points on the line that makes an angle of with the positive polar ais, as shown in Figure.9. To convert to rectangular form, make use of the relationship tan Polar equation Rectangular equation c. The graph of the polar equation r sec is not evident b simple inspection, so convert to rectangular form b using the relationship r cos. r sec Polar equation tan r tan Rectangular equation Now ou see that the graph is a vertical line, as shown in Figure.7. Now tr Eercise 9. r cos. FIGURE.7

57 Section.7 Polar Coordinates 78.7 EXERCISES VOCABULARY: Fill in the blanks.. The origin of the polar coordinate sstem is called the. See for worked-out solutions to odd-numbered eercises.. For the point r,, r is the from O to P and is the, counterclockwise from the polar ais to the line segment OP.. To plot the point r,, use the coordinate sstem.. The polar coordinates r, are related to the rectangular coordinates, as follows: tan r SKILLS AND APPLICATIONS In Eercises 5 8, plot the point given in polar coordinates and find two additional polar representations of the point, using < <. 5 5.,. 7., 8. 9.,..,.,., 7., 7 5.,..,.7 7., ,. In Eercises 9 8, a point in polar coordinates is given. Convert the point to rectangular coordinates. ( r, θ) =,., 5., 9.,., ( ) ( ) (r, θ) =, 5.,., 5 5,, 5, ( r, θ) =, ( ) (r, θ) = (, ) 5., 7., ,. 8., 5.7 In Eercises 9, use a graphing utilit to find the rectangular coordinates of the point given in polar coordinates. Round our results to two decimal places. 9., 9., 9..5,.. 8.5,.5..5, , ,.5. 8.,. In Eercises 7 5, a point in rectangular coordinates is given. Convert the point to polar coordinates. 7., 8., 9.,.,.,.,., 5., 5 5.,., 7., 8., 9., 5., 5., 9 5., 5. 5, 5. 7, 5 In Eercises 55, use a graphing utilit to find one set of polar coordinates for the point given in rectangular coordinates. 55., 5., 57. 5, 58. 7, 59.,. 5,.. 9 5,.. 5, 7, 7 9, In Eercises 5 8, convert the rectangular equation to polar form. Assume a >

58 78 Chapter Topics in Analtic Geometr a a 8. 9a 8. a 8. a In Eercises 85 8, convert the polar equation to rectangular form. 85. r sin 8. r cos 87. r cos 88. r 5 sin r 9. r 95. r csc 9. r csc 97. r sec 98. r sec 99. r cos. r sin. r sin. r cos. r sin. r cos 5. r. r sin cos 7. r 8. r sin cos sin In Eercises 9 8, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. 9. r. r 8... r sin. r cos 5. r cos. r sin 7. r sec 8. r csc EXPLORATION 5 5 TRUE OR FALSE? In Eercises 9 and, determine whether the statement is true or false. Justif our answer. 9. If n for some integer n, then r, and r, represent the same point on the polar coordinate sstem.. If r r, then r, and r, represent the same point on the polar coordinate sstem.. Convert the polar equation r h cos k sin to rectangular form and verif that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.. Convert the polar equation r cos to rectangular form and identif the graph.. THINK ABOUT IT (a) Show that the distance between the points r, and r is r r r r cos,. (b) Describe the positions of the points relative to each other for. Simplif the Distance Formula for this case. Is the simplification what ou epected? Eplain. (c) Simplif the Distance Formula for 9. Is the simplification what ou epected? Eplain. (d) Choose two points on the polar coordinate sstem and find the distance between them. Then choose different polar representations of the same two points and appl the Distance Formula again. Discuss the result.. GRAPHICAL REASONING (a) Set the window format of our graphing utilit on rectangular coordinates and locate the cursor at an position off the coordinate aes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Eplain the changes in the coordinates. Now repeat the process moving the cursor verticall. (b) Set the window format of our graphing utilit on polar coordinates and locate the cursor at an position off the coordinate aes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Eplain the changes in the coordinates. Now repeat the process moving the cursor verticall. (c) Eplain wh the results of parts (a) and (b) are not the same. 5. GRAPHICAL REASONING sin (a) Use a graphing utilit in polar mode to graph the equation r. (b) Use the trace feature to move the cursor around the circle. Can ou locate the point, 5? (c) Can ou find other polar representations of the point, 5? If so, eplain how ou did it.. CAPSTONE In the rectangular coordinate sstem, each point, has a unique representation. Eplain wh this is not true for a point r, in the polar coordinate sstem.

59 Section.8 Graphs of Polar Equations 78.8 GRAPHS OF POLAR EQUATIONS What ou should learn Graph polar equations b point plotting. Use smmetr to sketch graphs of polar equations. Use zeros and maimum r-values to sketch graphs of polar equations. Recognize special polar graphs. Wh ou should learn it Equations of several common figures are simpler in polar form than in rectangular form. For instance, Eercise on page 789 shows the graph of a circle and its polar equation. Introduction In previous chapters, ou learned how to sketch graphs on rectangular coordinate sstems. You began with the basic point-plotting method. Then ou used sketching aids such as smmetr, intercepts, asmptotes, periods, and shifts to further investigate the natures of graphs. This section approaches curve sketching on the polar coordinate sstem similarl, beginning with a demonstration of point plotting. Eample Graphing a Polar Equation b Point Plotting Sketch the graph of the polar equation r sin. Solution The sine function is periodic, so ou can get a full range of r-values b considering values of in the interval, as shown in the following table. 5 7 r If ou plot these points as shown in Figure.7, it appears that the graph is a circle of radius whose center is at the point,,. Circle: r = sin θ FIGURE.7 Now tr Eercise 7. You can confirm the graph in Figure.7 b converting the polar equation to rectangular form and then sketching the graph of the rectangular equation. You can also use a graphing utilit set to polar mode and graph the polar equation or set the graphing utilit to parametric mode and graph a parametric representation.

60 78 Chapter Topics in Analtic Geometr Smmetr In Figure.7 on the preceding page, note that as increases from to the graph is traced out twice. Moreover, note that the graph is smmetric with respect to the line Had ou known about this smmetr and retracing ahead of time, ou could have used fewer points. Smmetr with respect to the line is one of three important tpes of smmetr to consider in polar curve sketching. (See Figure.7.). ( r, θ) (r, θ) θ θ (r, θ) Smmetr with Respect to the Line FIGURE.7 θ θ (r, θ) (r, θ) ( r, θ) Smmetr with Respect to the Polar Ais + θ θ ( r, θ) (r, + θ) (r, θ) Smmetr with Respect to the Pole Note in Eample that cos cos. This is because the cosine function is even. Recall from Section. that the cosine function is even and the sine function is odd. That is, sin sin. Tests for Smmetr in Polar Coordinates The graph of a polar equation is smmetric with respect to the following if the given substitution ields an equivalent equation.. The line Replace r, b r, or r,. :. The polar ais: Replace r, b r, or r,.. The pole: Replace r, b r, or r,. Eample Using Smmetr to Sketch a Polar Graph Use smmetr to sketch the graph of r cos. r = + cos θ 5 Solution Replacing r, b r, produces r cos cos. cos cos So, ou can conclude that the curve is smmetric with respect to the polar ais. Plotting the points in the table and using polar ais smmetr, ou obtain the graph shown in Figure.7. This graph is called a limaçon. FIGURE.7 r 5 Now tr Eercise.

61 Section.8 Graphs of Polar Equations Spiral of Archimedes: r = θ +, θ FIGURE.7 The three tests for smmetr in polar coordinates listed on page 78 are sufficient to guarantee smmetr, but the are not necessar. For instance, Figure.7 shows the graph of r to be smmetric with respect to the line and et the tests on page 78 fail to indicate smmetr because neither of the following replacements ields an equivalent equation. Original Equation Replacement New Equation r r The equations discussed in Eamples and are of the form r sin f sin r, b r, r, b r, and, r r r cos gcos., The graph of the first equation is smmetric with respect to the line and the graph of the second equation is smmetric with respect to the polar ais. This observation can be generalized to ield the following tests. Quick Tests for Smmetr in Polar Coordinates. The graph of r f sin is smmetric with respect to the line. The graph of r gcos is smmetric with respect to the polar ais.. Zeros and Maimum r-values Two additional aids to graphing of polar equations involve knowing the -values for which r is maimum and knowing the -values for which r. For instance, in Eample, the maimum value of r for r sin is r, and this occurs when as shown in Figure.7. Moreover, r when,. Eample Sketching a Polar Graph Sketch the graph of r cos. Solution From the equation r cos, ou can obtain the following. 5 Smmetr: Maimum value of : r when Zero of r: r With respect to the polar ais r when The table shows several -values in the interval,. B plotting the corresponding points, ou can sketch the graph shown in Figure Limaçon: r = cos θ FIGURE.75 r.7.7 Note how the negative r-values determine the inner loop of the graph in Figure.75. This graph, like the one in Figure.7, is a limaçon. Now tr Eercise 5. 5

62 78 Chapter Topics in Analtic Geometr Some curves reach their zeros and maimum r-values at more than one point, as shown in Eample. Eample Sketching a Polar Graph Sketch the graph of r cos. Solution Smmetr: r With respect to the polar ais r,,, Maimum value of : when or Zeros of r: r when or,, 5,,, 5,, 5 r B plotting these points and using the specified smmetr, zeros, and maimum values, ou can obtain the graph shown in Figure.7. This graph is called a rose curve, and each of the loops on the graph is called a petal of the rose curve. Note how the entire curve is generated as increases from to. TECHNOLOGY Use a graphing utilit in polar mode to verif the graph of r cos shown in Figure.7. FIGURE.7 Now tr Eercise 9. 5

63 Section.8 Graphs of Polar Equations 787 Special Polar Graphs Several important tpes of graphs have equations that are simpler in polar form than in rectangular form. For eample, the circle r sin in Eample has the more complicated rectangular equation. Several other tpes of graphs that have simple polar equations are shown below. Limaçons r a ± b cos r a ± b sin a >, b > a a < a a b < b b < b Limaçon with Cardioid Dimpled Conve inner loop (heart-shaped) limaçon limaçon Rose Curves n petals if n is odd, n petals if n is even n. n = n = a a a a n = 5 n = r a cos n r a cos n r a sin n r a sin n Rose curve Rose curve Rose curve Rose curve Circles and Lemniscates a a a a r a cos r a sin r a sin r a cos Circle Circle Lemniscate Lemniscate

64 788 Chapter Topics in Analtic Geometr Eample 5 Sketching a Rose Curve, θ = θ = (, ) (, ) r = cos θ FIGURE.77 ( ) ( ), Sketch the graph of r cos. Solution Tpe of curve: Rose curve with n petals Smmetr: With respect to polar ais, the line and the pole, Maimum value of : when Zeros of r: r r r when,,,, Using this information together with the additional points shown in the following table, ou obtain the graph shown in Figure.77. r Now tr Eercise. Eample Sketching a Lemniscate Sketch the graph of r 9 sin. ( ), r = 9 sin θ FIGURE.78 ( ), Solution Tpe of curve: Smmetr: Lemniscate With respect to the pole Maimum value of : when Zeros of r: r when If sin <, this equation has no solution points. So, ou restrict the values of to those for which sin. r or r, Moreover, using smmetr, ou need to consider onl the first of these two intervals. B finding a few additional points (see table below), ou can obtain the graph shown in Figure r ±sin ± ± ± Now tr Eercise 7.

65 Section.8 Graphs of Polar Equations EXERCISES VOCABULARY: Fill in the blanks.. The graph of r f sin is smmetric with respect to the line.. The graph of r gcos is smmetric with respect to the.. The equation r cos represents a.. The equation r cos represents a. 5. The equation r sin represents a.. The equation r sin represents a. SKILLS AND APPLICATIONS See for worked-out solutions to odd-numbered eercises. In Eercises 7, identif the tpe of polar graph r = 5 cos θ 9.. r = ( cos θ ).. r = sin θ r = cosθ 8 5 r = 5 5 sin θ r = cos θ In Eercises 8, sketch the graph of the polar equation using smmetr, zeros, maimum r-values, and an other additional points.. r. r 7 5. r. 7. r sin r cos.. r sin.. r sin. r sin 5. r sin. r cos 7. r cos 8. r cos 9. r 5 sin. r cos. r cos. r sin. r sec. r 5 csc 5. r. r sin sin cos cos r r cos r sin r cos 7. r 9 cos 8. r sin 5 In Eercises 9 58, use a graphing utilit to graph the polar equation. Describe our viewing window. 9. r 9 5. r 5 In Eercises 8, test for smmetr with respect to the polar ais, and the pole. /,. r cos. r 9 cos 5. r. r sin cos 7. r cos 8. r 5 sin In Eercises 9, find the maimum value of zeros of r. 9. r sin. r cos. r cos. r sin r and an 5. r 5 5. r 8 5. r 8 cos 5. r cos 55. r sin 5. r cos 57. r 8 sin cos 58. r csc 5 In Eercises 59, use a graphing utilit to graph the polar equation. Find an interval for for which the graph is traced onl once. 59. r 8 cos. r 5 cos 5. r cos. r sin

66 79 Chapter Topics in Analtic Geometr. r. r sin In Eercises 5 8, use a graphing utilit to graph the polar equation and show that the indicated line is an asmptote of the graph. Name of Graph Polar Equation Asmptote 5. Conchoid. Conchoid 7. Hperbolic spiral 8. Strophoid EXPLORATION TRUE OR FALSE? In Eercises 9 and 7, determine whether the statement is true or false. Justif our answer. 9. In the polar coordinate sstem, if a graph that has smmetr with respect to the polar ais were folded on the line the portion of the graph above the polar ais would coincide with the portion of the graph below the polar ais. 7. In the polar coordinate sstem, if a graph that has smmetr with respect to the pole were folded on the line the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold. 7. Sketch the graph of r cos over each interval. Describe the part of the graph obtained in each case. (a) (c) (b) (d) 7. GRAPHICAL REASONING Use a graphing utilit to graph the polar equation r cos for (a) (b) and (c) Use the graphs to describe the effect of the angle. Write the equation as a function of sin for part (c). 7. The graph of r f is rotated about the pole through an angle. Show that the equation of the rotated graph is r f., 7. Consider the graph of r f sin. (a) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f cos. (b) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f sin.,,, r sec r csc r r cos sec. (c) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f cos. In Eercises 75 78, use the results of Eercises 7 and Write an equation for the limaçon r sin after it has been rotated through the given angle. (a) (b) (c) (d) 7. Write an equation for the rose curve r sin after it has been rotated through the given angle. (a) (b) (c) (d) 77. Sketch the graph of each equation. (a) (b) 78. Sketch the graph of each equation. (a) (c) r sin r sec r sec 79. THINK ABOUT IT How man petals do the rose curves given b r cos and r sin have? Determine the numbers of petals for the curves given b r cos n and r sin n, where n is a positive integer. 8. Use a graphing utilit to graph and identif r k sin for k,,, and. (b) (d) 8. Consider the equation r sin k. r sin r sec r sec (a) Use a graphing utilit to graph the equation for k.5. Find the interval for over which the graph is traced onl once. (b) Use a graphing utilit to graph the equation for k.5. Find the interval for over which the graph is traced onl once. (c) Is it possible to find an interval for over which the graph is traced onl once for an rational number k? Eplain. 8. CAPSTONE Write a brief paragraph that describes wh some polar curves have equations that are simpler in polar form than in rectangular form. Besides a circle, give an eample of a curve that is simpler in polar form than in rectangular form. Give an eample of a curve that is simpler in rectangular form than in polar form.

67 Section.9 Polar Equations of Conics 79.9 POLAR EQUATIONS OF CONICS What ou should learn Define conics in terms of eccentricit. Write and graph equations of conics in polar form. Use equations of conics in polar form to model real-life problems. Wh ou should learn it The orbits of planets and satellites can be modeled with polar equations. For instance, in Eercise 5 on page 79, a polar equation is used to model the orbit of a satellite. Alternative Definition of Conic In Sections. and., ou learned that the rectangular equations of ellipses and hperbolas take simple forms when the origin lies at their centers. As it happens, there are man important applications of conics in which it is more convenient to use one of the foci as the origin. In this section, ou will learn that polar equations of conics take simple forms if one of the foci lies at the pole. To begin, consider the following alternative definition of conic that uses the concept of eccentricit. Alternative Definition of Conic The locus of a point in the plane that moves so that its distance from a fied point (focus) is in a constant ratio to its distance from a fied line (directri) is a conic. The constant ratio is the eccentricit of the conic and is denoted b e. Moreover, the conic is an ellipse if e <, a parabola if e, and a hperbola if e >. (See Figure.79.) In Figure.79, note that for each tpe of conic, the focus is at the pole. Directri Q P Directri Directri Q P F = (, ) Q P F = (, ) P Q F = (, ) Corbis Ellipse: < e < Parabola: e Hperbola e > PF PF PF PF > PQ < PQ PQ FIGURE.79 PQ Polar Equations of Conics The benefit of locating a focus of a conic at the pole is that the equation of the conic takes on a simpler form. For a proof of the polar equations of conics, see Proofs in Mathematics on page 8. Polar Equations of Conics The graph of a polar equation of the form ep ep. r or. r ± e cos ± e sin is a conic, where e > is the eccentricit and is the distance between the focus (pole) and the directri. p