Chapter 9. Topics in Analytic Geometry. Selected Applications

Transcription

1 Chapter 9 Topics in Analtic Geometr 9. Circles and Parabolas 9. Ellipses 9. Hperbolas 9. Rotation and Sstems of Quadratic Equations 9.5 Parametric Equations 9. Polar Coordinates 9.7 Graphs of Polar Equations 9.8 Polar Equations of Conics Selected Applications Analtic geometr concepts have man real-life applications. The applications listed below represent a small sample of the applications in this chapter. Earthquake, Eercise 5, page 7 Suspension Bridge, Eercise 9, page 9 Architecture, Eercises 7 9, page 78 Satellite Orbit, Eercise 5, page 79 Navigation, Eercise, page 88 Projectile Motion, Eercises 55 58, page 70 Planetar Motion, Eercises 9 55, page 77 Sports, Eercises 95 98, page Conics are used to represent man real-life phenomena such as reflectors used in flashlights, orbits of planets, and navigation. In Chapter 9, ou will learn how to write and graph equations of conics in rectangular and polar coordinates. You will also learn how to graph other polar equations and curves represented b parametric equations. AP/Wide World Photos Satellites are used to monitor weather patterns, collect scientific data, and assist in navigation. Satellites orbit Earth in elliptical paths. 59

2 0 Chapter 9 Topics in Analtic Geometr 9. Circles and Parabolas Conics Conic sections were discovered during the classical Greek period, 00 to 00 B.C. The earl Greek studies were largel concerned with the geometric properties of conics. It was not until the earl 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 double-napped 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 9.. What ou should learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in standard form. Write equations of parabolas in standard form. 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 95 on page 9, a parabola is used to design the entrance ramp for a highwa. Circle Ellipse Parabola Hperbola Figure 9. Basic Conics Point Line Two intersecting lines Figure 9. Degenerate Conics Roalt-Free/Corbis 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 0. However, ou will stud a third approach, in which each of the conics is defined as a locus (collection) of points satisfing a certain geometric propert. For eample, the definition of a circle as the collection of all points (, ) that are equidistant from a fied point (h, k) leads to the standard equation of a circle h k r. Equation of circle

3 Section 9. Circles and Parabolas Circles The definition of a circle as a locus of points is a more general definition of a circle as it applies to conics. Definition of a Circle A circle is the set of all points, in a plane that are equidistant from a fied point h, k, called the center of the circle. (See Figure 9..) The distance r between the center and an point, on the circle is the radius. The Distance Formula can be used to obtain an equation of a circle whose center is h, k and whose radius is r. h k r Distance Formula h k r Square each side. Standard Form of the Equation of a Circle The standard form of the equation of a circle is h k r. The point h, k is the center of the circle, and the positive number r is the radius of the circle. The standard form of the equation of a circle whose center is the origin, h, k 0, 0, is r. Figure 9. (h, k) r (, ) Eample Finding the Standard Equation of a Circle The point, is on a circle whose center is at,, as shown in Figure 9.. Write the standard form of the equation of the circle. Solution The radius of the circle is the distance between, and,. r Use Distance Formula. 7 Simplif. 58 Simplif. The equation of the circle with center h, k, and radius r 58 is h k r Standard form 58 Substitute for h, k, and r. 58. Simplif. Now tr Eercise. (, ) 8 8 (, ) 8 0 Figure 9.

4 Chapter 9 Topics in Analtic Geometr Eample Sketching a Circle Sketch the circle given b the equation 0 and identif its center and radius. Solution Begin b writing the equation in standard form. 0 Write original equation. 9 9 Complete the squares. Write in standard form. In this form, ou can see that the graph is a circle whose center is the point, and whose radius is r. Plot several points that are two units from the center. The points 5,,,,,, and, are convenient. Draw a circle that passes through the four points, as shown in Figure 9.5. Now tr Eercise. Prerequisite Skills To use a graphing utilit to graph a circle, review Appendi B.. Figure 9.5 Eample Finding the Intercepts of a Circle Find the - and -intercepts of the graph of the circle given b the equation. Solution To find an -intercepts, let 0. To find an -intercepts, let 0. -intercepts: 0 ± ± Substitute 0 for. Simplif. Take square root of each side. Add to each side. -intercepts: 0 Substitute 0 for. 0 Simplif. 0 Take square root of each side. Add to each side. So the -intercepts are, 0 and, 0, and the -intercept is 0,, as shown in Figure 9.. Now tr Eercise 9. (0, ) ( +, 0) Figure 9. 8 ( ) + ( ) = (, 0)

5 Section 9. Circles and Parabolas 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 a Parabola Ais A parabola is the set of all points, in a plane that are equidistant from a fied line, the directri, and a fied point, the focus, not on the line. (See Figure 9.7.) The midpoint between the focus and the directri is the verte, and the line passing through the focus and the verte is the ais of the parabola. Directri Focus Verte d d d (, ) d Note in Figure 9.7 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. Figure 9.7 Standard Equation of a Parabola (See the proof on page 77.) The standard form of the equation of a parabola with verte at h, k is as follows. h p k, p 0 Vertical ais; directri: k p k p h, p 0 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 0, 0, the equation takes one of the following forms. p Vertical ais p Horizontal ais See Figure 9.8. Focus: ( h, k + p) Verte: ( h, k) Ais: = h p > 0 Directri: = k p Verte: (h, k) Ais: = h Directri: = k p p < 0 Focus: (h, k + p) Directri: = h p p > 0 Verte: ( h, k) Focus: ( h + p, k) Ais: = k p < 0 Focus: (h + p, k) Directri: = h p Ais: = k Verte: (h, k) h p k h p k k p h k p h (a) Vertical ais: p > 0 (b) Vertical ais: p < 0 (c) Horizontal ais: p > 0 (d) Horizontal ais: p < 0 Figure 9.8

6 Chapter 9 Topics in Analtic Geometr Eample Finding the Standard Equation of a Parabola Find the standard form of the equation of the parabola with verte at the origin and focus 0,. Solution Because the ais of the parabola is vertical, passing through 0, 0 and 0,, consider the equation p. Because the focus is p units from the verte, the equation is. You can obtain the more common quadratic form as follows. Write original equation. Multipl each side b. Use a graphing utilit to confirm that the graph is a parabola, as shown in Figure 9.9. Now tr Eercise 5. = 0 Focus: (0, ) 9 9 Figure 9.9 Verte: (0, 0) Eample 5 Finding the Focus of a Parabola Find the focus of the parabola given b. Solution To find the focus, convert to standard form b completing the square. Comparing this equation with h p k Write original equation. Multipl each side b. Add to each side. Complete the square. Combine like terms. Write in standard form. ou can conclude that h, k, and p. Because p is negative, the parabola opens downward, as shown in Figure 9.0. Therefore, the focus of the parabola is h, k p,. Now tr Eercise. Focus = + Verte: (, ) Focus: (, Figure 9.0 (

7 Section 9. Circles and Parabolas 5 Eample Finding the Standard Equation of a Parabola Find the standard form of the equation of the parabola with verte, 0 and focus at, 0. Solution Because the ais of the parabola is horizontal, passing through, 0 and, 0, consider the equation k p h where h, k 0, and p. So, the standard form is 0. The parabola is shown in Figure 9.. = ( ) Focus: (, 0) 5 Figure 9. Now tr Eercise 77. Verte: (, 0) TECHNOLOGY TIP Use a graphing utilit to confirm the equation found in Eample. To do this, it helps to graph the equation using two separate equations: (upper part) and (lower part). Note that when ou graph conics using two separate equations, our graphing utilit ma not connect the two parts. This is because some graphing utilities are limited in their resolution. So, in this tet, a blue curve is placed behind the graphing utilit s displa to indicate where the graph should appear. Reflective Propert of Parabolas 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 about 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 9.. 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. Light source at focus Figure 9. Focus Ais Parabolic reflector: Light is reflected in parallel ras.

8 Chapter 9 Topics in Analtic Geometr Reflective Propert of a Parabola Ais The tangent line to a parabola at a point P makes equal angles with the following two lines (see Figure 9.).. The line passing through P and the focus Focus α P Eample 7. The ais of the parabola Finding the Tangent Line at a Point on a Parabola α Tangent line Find the equation of the tangent line to the parabola given b at the point,. Figure 9. Solution For this parabola, p and the focus is 0,, as shown in Figure 9.. You can find the -intercept 0, b of the tangent line b equating the lengths of the two sides of the isosceles triangle shown in Figure 9.: d b = ( ) 0, d α (, ) and d 0 5. d α 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. 0 and the equation of the tangent line in slope-intercept form is. Now tr Eercise 85. Figure 9. (0, b) TECHNOLOGY TIP Tr using a graphing utilit to confirm the result of Eample 7. B graphing and in the same viewing window, ou should be able to see that the line touches the parabola at the point,.

9 Section 9. Circles and Parabolas 7 9. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. A is the intersection of a plane and a double-napped cone.. A collection of points satisfing a geometric propert can also be referred to as a of points.. A is the set of all points, in a plane that are equidistant from a fied point, called the.. A is 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 of a parabola is the midpoint between the focus and the directri.. The line that passes through the focus and verte of a parabola is called the of the parabola. 7. 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. In Eercises, find the standard form of the equation of the circle with the given characteristics.. Center at origin; radius: 8. Center at origin; radius:. Center:, 7; point on circle:, 0. Center:, ; point on circle:, 5. Center:, ; diameter: 7. Center: 5, ; diameter: In Eercises 7, identif the center and radius of the circle In Eercises 0, write the equation of the circle in standard form. Then identif its center and radius In Eercises 8, sketch the circle. Identif its center and radius In Eercises 9, find the - and -intercepts of the graph of the circle Earthquake An earthquake was felt up to 8 miles from its epicenter. You were located 0 miles west and 5 miles south of the epicenter. (a) Let the epicenter be at the point 0, 0. Find the standard equation that describes the outer boundar of the earthquake. (b) Would ou have felt the earthquake? (c) Verif our answer to part (b) b graphing the equation of the outer boundar of the earthquake and plotting our location. How far were ou from the outer boundar of the earthquake?. Landscaper A landscaper has installed a circular sprinkler sstem that covers an area of 800 square feet. (a) Find the radius of the region covered b the sprinkler sstem. Round our answer to three decimal places. (b) If the landscaper wants to cover an area of 00 square feet, how much longer does the radius need to be?

10 8 Chapter 9 Topics in Analtic Geometr In Eercises 7, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f ).] (a) (b ) (c) 8 7 (d) (e) 8 (f) In Eercises 7 8, find the standard form of the equation of the parabola with the given characteristics (, ) (.5, ) 9 (, 0) (5, ) (, 0) In Eercises 5, find the standard form of the equation of the parabola with the given characteristic(s) and verte at the origin (, ) (, ) , 0 5. Focus:. Focus: 7. Focus:, 0 8. Focus: 0, 9. Directri: 50. Directri: 5. Directri: 5. Directri: 5. Horizontal ais and passes through the point, 5. Vertical ais and passes through the point, In Eercises 55 7, find the verte, focus, and directri of the parabola and sketch its graph Verte:, 0; focus: 7. Verte:, ; focus:, Verte: 5, ; focus:, 78. Verte:, ; focus:, Verte: 0, ; directri: 80. Verte:, ; directri: 8. Focus:, ; directri: 8. Focus: 0, 0; directri: In Eercises 8 and 8, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utilit to graph both in the same viewing window. Determine the coordinates of the point of tangenc. Parabola Tangent Line In Eercises 85 88, find an equation of the tangent line to the parabola at the given point and find the -intercept of the line ,, 8,, 9,,,, 8, Revenue The revenue R (in dollars) generated b the sale of -inch televisions is modeled b R 75. Use a graphing utilit to graph the function and approimate the sales that will maimize revenue.

11 Section 9. Circles and Parabolas Beam Deflection A simpl supported beam is feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is inch. The shape of the deflected beam is parabolic. ft in. (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 (a) Find an equation of the parabola. (Assume that the origin is at the center of the beam.) (b) How far from the center of the beam is the deflection equal to inch? Not drawn to scale 9. Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is feet wide is 0. foot higher in the center than it is on the sides (see figure). 9. Automobile Headlight The filament of an automobile headlight is at the focus of a parabolic reflector, which sends light out in a straight beam (see figure). 8 in. ft 0. ft Not drawn to scale.5 in. (a) The filament of the headlight is.5 inches from the verte. Find an equation for the cross section of the reflector. (b) The reflector is 8 inches wide. Find the depth of the reflector. 9. Solar Cooker You want to make a solar hot dog cooker using aluminum foil-lined cardboard, shaped as a parabolic trough. The figure shows how to suspend the hot dog with a wire through the foci of the ends of the parabolic trough. The parabolic end pieces are inches wide and inches deep. How far from the bottom of the trough should the wire be inserted? in. (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 0. foot lower than in the middle? 95. Highwa 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. 800 Interstate (000, 800) 00 in. 9. Suspension Bridge Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 80 meters apart. The top of each tower is 5 meters above the roadwa. The cables touch the roadwa midwa between the towers (000, 800) Street

12 70 Chapter 9 Topics in Analtic Geometr 9. Satellite Orbit A satellite in a 00-mile-high circular orbit around Earth has a velocit of approimatel 7,500 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). (a) Find the escape velocit of the satellite. (b) Find an equation of its path (assume the radius of Earth is 000 miles). 97. Path of a Projectile The path of a softball is modeled b The coordinates and are measured in feet, with 0 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 zoom and trace features of the graphing utilit to approimate the highest point the ball reaches and the distance the ball travels. 98. Projectile Motion 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 v s. In this model, air resistance is disregarded, is the height (in feet) of the projectile, and is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 75-foot tower with a velocit of feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontall before striking the ground? In Eercises 99 0, find an equation of the tangent line to the circle at the indicated point. Recall from geometr that the tangent line to a circle is perpendicular to the radius of the circle at the point of tangenc. 99. Circle 5 00 miles Circular orbit Point, Parabolic path Not drawn to scale Circle Snthesis Point True or False? In Eercises 0 08, determine whether the statement is true or false. Justif our answer. 0. The equation 5 5 represents a circle with its center at the origin and a radius of The graph of the equation r will have -intercepts ±r, 0 and -intercepts 0, ±r. 05. A circle is a degenerate conic. 0. It is possible for a parabola to intersect its directri. 07. The point which lies on the graph of a parabola closest to its focus is the verte of the parabola. 08. The directri of the parabola intersects, or is tangent to, the graph of the parabola at its verte, 0, Writing Cross sections of television antenna dishes are parabolic in shape (see figure). Write a paragraph describing wh these dishes are parabolic. Include a graphical representation of our description. 0. Think About It The equation 0 is a degenerate conic. Sketch the graph of this equation and identif the degenerate conic. Describe the intersection of the plane with the double-napped cone for this particular conic. Think About It In Eercises and, change the equation so that its graph matches the description.. ; upper half of parabola. ; lower half of parabola Skills Review 5,, 5, Amplifier Dish reflector Cable to radio or TV In Eercises, use a graphing utilit to approimate an relative minimum or maimum values of the function.. f. f 5. f. f 5

13 Section 9. Ellipses 7 9. Ellipses Introduction The third tpe of conic is called an ellipse. It is defined as follows. Definition of an 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 9.5(a).] Focus d (, ) d Focus Major ais Verte Center Minor ais Verte What ou should learn Write equations of ellipses in standard form. 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 50 on page 78, an ellipse is used to model the floor of Statuar Hall, an elliptical room in the U.S. Capitol Building in Washington, D.C. d + d is constant. (a) Figure 9.5 (b) 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. [See Figure 9.5(b).] You can visualize the definition of an ellipse b imagining two thumbtacks placed at the foci, as shown in Figure 9.. 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. John Neubauer/PhotoEdit (, ) b + c b + c ( h, k) b Figure 9. To derive the standard form of the equation of an ellipse, consider the ellipse in Figure 9.7 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 a c a c a or simpl the length of the major ais. Length of major ais b + c = a b + c = a Figure 9.7 c a

14 7 Chapter 9 Topics in Analtic Geometr Now, if ou let, be an point on the ellipse, the sum of the distances between, and the two foci must also be a. That is, h c k h c k a. Finall, in Figure 9.7, 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. 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 0 < b < a, is h a h 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 0, 0, the equation takes one of the following forms. a b b a k b k a. Major ais is horizontal. Major ais is vertical. Eploration On page 7 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. Tr doing this. Var the length of the string and the distance between the thumbtacks. Eplain how to obtain ellipses that are almost circular. Eplain how to obtain ellipses that are long and narrow. Figure 9.8 shows both the vertical and horizontal orientations for an ellipse. ( h) ( k) + = a b ( h) ( k) + = b a ( h, k) b ( h, k) a a b Major ais is horizontal. Figure 9.8 Major ais is vertical.

15 Section 9. Ellipses 7 Eample Finding the Standard Equation of an Ellipse Find the standard form of the equation of the ellipse having foci at 0, and, and a major ais of length, as shown in Figure 9.9. Solution B the Midpoint Formula, 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 9 5. Because the major ais is horizontal, the standard equation is 5. Now tr Eercise 5. (0, ) b = 5 (, ) a = Figure 9.9 (, ) TECHNOLOGY SUPPORT For instructions on how to use the zoom and trace features, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Eample Sketching an Ellipse Sketch the ellipse given b and identif the center and vertices. Algebraic Solution Write original equation. Divide each side b. Write in standard form. The center of the ellipse is 0, 0. Because the denominator of the -term is larger than the denominator of the -term, ou can conclude that the major ais is vertical. Moreover, because a, the vertices are 0, and 0,. Finall, because b, the endpoints of the minor ais are, 0 and, 0, as shown in Figure 9.0. Graphical Solution Solve the equation of the ellipse 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 9., ou can see that the major ais is vertical and its center is at the point 0, 0. You can use the zoom and trace features to approimate the vertices to be 0, and 0,. 8 = 8 = Figure 9. Figure 9.0 Now tr Eercise.

16 7 Chapter 9 Topics in Analtic Geometr Eample Graphing an Ellipse Graph the ellipse given b Solution Begin b writing the original equation in standard form. In the third step, note that 9 and are added to both sides of the equation when completing the squares Write original equation. Group terms and factor out of -terms. Write in standard form. Now ou see 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. The graph of this ellipse is shown in Figure 9.. Eample Now tr Eercise 5. Analzing an Ellipse Find the center, vertices, and foci of the ellipse Solution B completing the square, ou can write the original equation in standard form Write original equation. Group terms and factor out of -terms. 8 Write in completed square form. Write in standard form. So, the major ais is vertical, where h, k, a, b, and c a b. Therefore, ou have the following. Center:, Vertices:, Foci:,,, The graph of the ellipse is shown in Figure 9.. Now tr Eercise 7. Write in completed square form. ( + ) ( ) + = ( 5, ) (, ) (, ) (, ) (, 0) 0 Figure 9. ( ) ( + ) + = Figure 9. Verte Focus Center Focus Verte TECHNOLOGY TIP 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 obtain and. Use a viewing window in which 0 and. You should obtain the graph shown in Figure 9.. 9

17 Section 9. Ellipses 75 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 5 investigates the elliptical orbit of the moon about Earth. Eample 5 An Application Involving an Elliptical Orbit The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 9.. The major and minor aes of the orbit have lengths of 78,800 kilometers and 77,0 kilometers, respectivel. Find the greatest and smallest distances (the apogee and perigee) from Earth s center to the moon s center. 77,0 km Earth 78,800 km Moon STUDY TIP Note in Eample 5 and Figure 9. that Earth is not the center of the moon s orbit. Perigee Figure 9. Apogee Solution Because a 78,800 and b 77,0, ou have a 8,00 which implies that c a b and b 8,80 8,00 8,80,08. So, the greatest distance between the center of Earth and the center of the moon is a c 8,00,08 05,508 kilometers and the smallest distance is a c 8,00,08,9 kilometers. Now tr Eercise 5.

18 7 Chapter 9 Topics in Analtic Geometr 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 0 < e < for ever ellipse. 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 0 < c < a. For an ellipse that is nearl circular, the foci are close to the center and the ratio ca is small [see Figure 9.5(a)]. On the other hand, for an elongated ellipse, the foci are close to the vertices and the ratio ca is close to [see Figure 9.5(b)]. Foci Foci e = c a c e is small. e = c a c e is close to. a a (a) Figure 9.5 (b) The orbit of the moon has an eccentricit of e 0.059, and the eccentricities of the eight planetar orbits are as follows. Mercur: e 0.05 Jupiter: e 0.08 Venus: e Saturn: e 0.05 Earth: e 0.07 Uranus: e 0.07 Mars: e 0.09 Neptune: e 0.008

19 Section 9. Ellipses Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. An is the set of all points (, ) in a plane, the sum of whose distances from two distinct fied points 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 an ellipse is called the of the ellipse.. You can use the concept of to measure the ovalness of an ellipse. In Eercises, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (b) (d) (e) (f ) In Eercises 7, find the center, vertices, foci, and eccentricit of the ellipse, and sketch its graph. Use a graphing utilit to verif our graph In Eercises, (a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricit of the ellipse, (c) sketch the ellipse, and use a graphing utilit to verif our graph In Eercises 0, find the standard form of the equation of the ellipse with the given characteristics and center at the origin... (0, ) 0, (, 0) ) (, 0) (, 0) 9 9 (, 0) (0, ) 0, ) 5. Vertices: ±, 0; foci: ±, 0. Vertices: 0, ±8; foci: 0, ± 7. Foci: ±, 0; major ais of length 8 8. Foci: ±, 0; major ais of length 9. Vertices: 0, ±5; passes through the point, 0. Vertical major ais; passes through points 0, and, 0 ) )

20 78 Chapter 9 Topics in Analtic Geometr In Eercises 0, find the standard form of the equation of the ellipse with the given characteristics.. 7. (0, ) (, ) (, ) (, 0) (, ) 7 (, 0) (, ) 8 (, ). Vertices: 0,, 8, ; minor ais of length. Foci: 0, 0,, 0; major ais of length 5. Foci: 0, 0, 0, 8; major ais of length. Center:, ; verte:, ; minor ais of length 7. Vertices:,,, 9; minor ais of length 8. Center:, ; a c; foci:,, 5, 9. Center: 0, ; a c; vertices:,,, 0. Vertices: 5, 0, 5, ; endpoints of the minor ais: 0,, 0, In Eercises, find the eccentricit of the ellipse Find an equation of the ellipse with vertices ±5, 0 and eccentricit e 5.. Find an equation of the ellipse with vertices 0, ±8 and eccentricit e. 7. Architecture A semielliptical arch over a tunnel for a road through a mountain has a major ais of 00 feet and a height at the center of 0 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 over the tunnel. (c) Determine the height of the arch 5 feet from the edge of the tunnel. 8. Architecture A semielliptical arch through a railroad underpass has a major ais of feet and a height at the center of feet. (a) Draw a rectangular coordinate sstem on a sketch of the underpass with the center of the road entering the underpass at the origin. Identif the coordinates of the known points. (b) Find an equation of the semielliptical arch over the underpass. (c) Will a truck that is 0 feet wide and 9 feet tall be able to drive through the underpass without crossing the center line? Eplain our reasoning. 9. 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 on the wall b the method discussed on page 7. Give the required positions of the tacks and the length of the string. 50. Statuar Hall Statuar Hall is an elliptical room in the United States Capitol Building in Washington, D.C. The room is also referred to as the Whispering Galler because a person standing at one focus of the room can hear even a whisper spoken b a person standing at the other focus. Given that the dimensions of Statuar Hall are feet wide b 97 feet long, find an equation for the shape of the floor surface of the hall. Determine the distance between the foci. 5. Geometr The area of the ellipse in the figure is twice the area of the circle. What is the length of the major ais? Hint: The area of an ellipse is given b A ab. (0, 0) ( a, 0) (a, 0) (0, 0) 5. Astronom Halle s comet has an elliptical orbit with the sun at one focus. The eccentricit of the orbit is approimatel The length of the major ais of the orbit is about 5.88 astronomical units. (An astronomical unit is about 9 million miles.) Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major ais on the -ais. 5. Astronom The comet Encke has an elliptical orbit with the sun at one focus. Encke s orbit ranges from 0. to.08 astronomical units from the sun. Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major ais on the -ais.

21 Section 9. Ellipses 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. The center of Earth was a focus of the elliptical orbit, and the radius of Earth is 78 kilometers (see figure). Find the eccentricit of the orbit. 55. Geometr A line segment through a focus with endpoints on an ellipse, 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 this information ields other points on the curve (see figure). Show that the length of each latus rectum is b a. In Eercises 5 59, sketch the ellipse using the latera recta (see Eercise 55). 0. Writing Write an equation of an ellipse in standard form and graph it on paper. Do not write the equation on our graph. Echange graphs with another student. Use the graph ou receive to reconstruct the equation of the ellipse it represents and find its eccentricit. Compare our results and write a short paragraph discussing our findings. Snthesis Focus 8 km 97 km True or False? In Eercises and, determine whether the statement is true or false. Justif our answer. F F Latera recta. 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 ).. The area of a circle with diameter d r 8 is greater than the area of an ellipse with major ais a 8.. 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 (see Figure 9.). 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.. Eploration Consider the ellipse 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 a maimum area. (d) Use a graphing utilit to graph the area function to support our conjecture in part (c). 5. 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 0, is.. Proof Show that a b c for the ellipse a b where a > 0, b > 0, and the distance from the center of the ellipse 0, 0 to a focus is c. Skills Review In Eercises 7 70, determine whether the sequence is arithmetic, geometric, or neither. 7., 55,,,, , 0, 0, 0, 5,... 9.,,,,, ,,, 5, 7,... In Eercises 7 7, find the sum. n n n n a b 0. a A n n n0 n

22 80 Chapter 9 Topics in Analtic Geometr 9. Hperbolas Introduction 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 constant; whereas for a hperbola, the difference of the distances between the foci and a point on the hperbola is constant. Definition of a Hperbola A hperbola is the set of all points, in a plane, the difference of whose distances from two distinct fied points, the foci, is a positive constant. [See Figure 9.(a).] (, ) d d Focus Focus d d is a positive constant. (a) Figure 9. The graph of a hperbola has two disconnected parts called the branches. The line through the two foci intersects the hperbola at two points called the 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 9.(b)]. The development of the standard form of the equation of a hperbola is similar to that of an ellipse. Note that a, b, and c are related differentl for hperbolas than for ellipses. For a hperbola, the distance between the foci and the center is greater than the distance between the vertices and the center. c (b) a Transverse ais Branch Branch Verte Center Ve rte 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 on page 88, hperbolas are used to locate the position of an eplosion that was recorded b three listening stations. James Foote/Photo Researchers, Inc. Standard Equation of a Hperbola The standard form of the equation of a hperbola with center at h, k is h k Transverse ais is horizontal. a b k h. Transverse ais is vertical. a b 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 0, 0, the equation takes one of the following forms. a b Transverse ais is horizontal. a b Transverse ais is vertical.

23 Figure 9.7 shows both the horizontal and vertical orientations for a hperbola. Section 9. Hperbolas 8 ( h) ( k) = a b ( k) ( h) = a b ( h, k + c) ( h c, k) ( h, k) ( h + c, k) Transverse ais ( h, k) Transverse ais Transverse ais is horizontal. Figure 9.7 ( h, k c) Transverse ais is vertical. Eample Finding the Standard Equation of a Hperbola Find the standard form of the equation of the hperbola with foci, and 5, and vertices 0, and,. Solution B the Midpoint Formula, the center of the hperbola occurs at the point,. Furthermore, c 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 of the hperbola is. 5 Figure 9.8 shows the hperbola. ( ) ( ) ( 5 = (0, ) (5, ) (, ) (, ) 8 ( TECHNOLOGY TIP When using a graphing utilit to graph an equation, ou must solve the equation for before entering it into the graphing utilit. When graphing equations of conics, it can be difficult to solve for, which is wh it is ver important to know the algebra used to solve equations for. Figure 9.8 Now tr Eercise.

24 8 Chapter 9 Topics in Analtic Geometr Asmptotes of a Hperbola Each hperbola has two asmptotes that intersect at the center of the hperbola. The asmptotes pass through the corners of a rectangle of dimensions a b b, with its center at h, k, as shown in Figure 9.9. Conjugate ais ( h, k + b) Asmptote Asmptotes of a Hperbola k ± b Asmptotes h a Asmptotes for horizontal transverse ais k ± a h b for vertical transverse ais ( h, k) The conjugate ais of a hperbola is the line segment of length b joining h, k b and h, k b if the transverse ais is horizontal, and the line segment of length b joining h b, k and h b, k if the transverse ais is vertical. ( h, k b) ( h a, k) ( h + a, k) Figure 9.9 Asmptote Eample Sketching a Hperbola Sketch the hperbola whose equation is. Algebraic Solution Write original equation. Divide each side b. Write in standard form. Because the -term is positive, ou can conclude that the transverse ais is horizontal. So, the vertices occur at, 0 and, 0, the endpoints of the conjugate ais occur at 0, and 0,, and ou can sketch the rectangle shown in Figure 9.0. Finall, b drawing the asmptotes, and, through the corners of this rectangle, ou can complete the sketch, as shown in Figure 9.. 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 9., ou can see that the transverse ais is horizontal. You can use the zoom and trace features to approimate the vertices to be, 0 and, 0. = 9 9 = Figure 9. Figure 9.0 Figure 9. Now tr Eercise 5.

25 Section 9. Hperbolas 8 Eample Finding the Asmptotes of a Hperbola Sketch the hperbola given b 8 0 and find the equations of its asmptotes. Solution 8 0 Write original equation. Subtract from each side and factor. Complete the square. Write in completed square form. Write in standard form. From this equation ou can conclude that the hperbola has a vertical transverse ais, is centered at, 0, has vertices, and,, and has a conjugate ais with endpoints, 0 and, 0. To sketch the hperbola, draw a rectangle through these four points. The asmptotes are the lines passing through the corners of the rectangle, as shown in Figure 9.. Finall, using a and b, ou can conclude that the equations of the asmptotes are Figure 9. and. Now tr Eercise 9. TECHNOLOGY TIP 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 obtain and. Use a viewing window in which 9 9 and. You should obtain the graph shown in Figure 9.. 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. = + Figure 9. ( + ) 9 9 = + ( + )

26 8 Chapter 9 Topics in Analtic Geometr Eample Using Asmptotes to Find the Standard Equation Find the standard form of the equation of the hperbola having vertices, 5 and, and having asmptotes 8 and as shown in Figure 9.5. 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 9.5 = 8 (, ) 0 (, 5) m a b and m a b and because a, ou can conclude that b. So, the standard form of the equation is. Now tr Eercise 9. 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 9.(a). If the eccentricit is close to, the branches of the hperbola are more pointed, as shown in Figure 9.(b). e is large. e is close to. Verte Focus Verte Focus e = c a c e = c a a c a (a) Figure 9. (b)

27 Section 9. Hperbolas 85 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? Solution Assuming sound travels at 00 feet per second, ou know that the eplosion took place 00 feet farther from B than from A, as shown in Figure 9.7. The locus of all points that are 00 feet closer to A than to B is one branch of the hperbola where a b 00 B 000 A c and a So, b c a ,759,00, and ou can conclude that the eplosion occurred somewhere on the right branch of the hperbola 00 c a c a c = ( c a) = 580,0,000. 5,759,00 Figure 9.7 Now tr Eercise. Another interesting application of conic sections involves the orbits of comets in our solar sstem. Of the 0 comets identified prior to 970, 5 have elliptical orbits, 95 have parabolic orbits, and 70 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 9.8. Undoubtedl, there are man comets with parabolic or hperbolic orbits that have not been identified. You get to see such comets onl 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: v < GMp. Parabola: v GMp. Hperbola: v > GMp In each of these equations, M kilograms (the mass of the sun) and G.7 0 cubic meter per kilogram-second squared (the universal gravitational constant). Hperbolic orbit Verte Elliptical orbit Sun p Parabolic orbit Figure 9.8

28 8 Chapter 9 Topics in Analtic Geometr General Equations of Conics Classifing a Conic from Its General Equation The graph of A B C D E F 0 is one of the following.. Circle: A C A 0. Parabola: AC 0 A 0 or C 0, but not both.. Ellipse: AC > 0 A and C have like signs.. Hperbola: AC < 0 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 graphs are not conics. Eample Classifing Conics from General Equations Classif the graph of each equation. a b. 8 0 c. 0 d. 8 0 Solution a. For the equation 9 5 0, ou have AC 0 0. Parabola So, the graph is a parabola. b. For the equation 8 0, ou have AC < 0. Hperbola So, the graph is a hperbola. STUDY TIP Notice in Eample (a) that there is no -term in the equation. Therefore, C 0. c. For the equation 0, ou have AC > 0. Ellipse So, the graph is an ellipse. d. For the equation 8 0, ou have A C. Circle So, the graph is a circle. Now tr Eercise 9.

29 Section 9. Hperbolas Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. A is the set of all points, in a plane, the difference of whose distances from two distinct fied points 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. 5. The general form of the equation of a conic is given b. In Eercises, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) 8 In Eercises 5, find the center, vertices, foci, and asmptotes of the hperbola, and sketch its graph using the asmptotes as an aid. Use a graphing utilit to verif our graph. (b) (d) In Eercises 5, (a) find the standard form of the equation of the hperbola, (b) find the center, vertices, foci, and asmptotes of the hperbola, (c) sketch the hperbola, and use a graphing utilit to verif our graph In Eercises 5 0, find the standard form of the equation of the hperbola with the given characteristics and center at the origin. 5. Vertices: 0, ±; foci: 0, ±. Vertices: ±, 0; foci: ±, 0 7. Vertices: ±, 0; asmptotes: ±5 8. Vertices: 0, ±; asmptotes: ± 9. Foci: 0, ±8; asmptotes: ± 0. Foci: ±0, 0; asmptotes: ±

30 88 Chapter 9 Topics in Analtic Geometr In Eercises, find the standard form of the equation of the hperbola with the given characteristics.. Vertices:, 0,, 0; foci: 0, 0, 8, 0. Vertices:,,, ; foci:, 5,, 5. Vertices:,,, 9; foci:, 0,, 0. Vertices:,,, ); foci:,,, 5. Vertices:,,, ; passes through the point 0, 5. Vertices:,,, ; passes through the point 5, 7. Vertices: 0,, 0, 0; passes through the point 5, 8. Vertices:,,, ; passes through the point 0, 5 9. Vertices:,,, ; asmptotes:, 0. Vertices:, 0,, ; asmptotes:,. Vertices: 0,,, ; asmptotes:,. Vertices:, 0,, ; asmptotes:,. 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 00 feet per second.). Sound Location Three listening stations located at 00, 0, 00, 00, and 00, 0 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 00 feet per second.) 5. 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.. Navigation 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 (8,000 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, 00 miles apart, are positioned on a rectangular coordinate sstem at coordinates 50, 0 and 50, 0, and that a ship is traveling on a hperbolic path with coordinates, 75 (see figure) Station Station Not drawn to scale (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 000 microseconds (0.00 second). (b) Determine the distance between the ship and station when the ship reaches the shore. (c) The captain of the ship wants to enter a ba located between the two stations. The ba is 0 miles from station. What should be the time difference between the pulses? (d) The ship is 0 miles offshore when the time difference in part (c) is obtained. What is the position of the ship? 7. Hperbolic 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, 0. Find the verte of the mirror if the mount at the top edge of the mirror has coordinates,. (, ) Ba (, 9) (, 0) (, 9) (, 0) 8 8 (, 0) (, 0) (, 9) (, 9)

31 Section 9. Hperbolas Panoramic Photo A panoramic photo can be taken using a hperbolic mirror. The camera is pointed toward the verte of the mirror and the camera s optical center is positioned at one focus of the mirror (see figure). An equation for the cross-section of the mirror is. 5 Find the distance from the camera s optical center to the mirror. In Eercises 9 58, classif the graph of the equation as a circle, a parabola, an ellipse, or a hperbola. Snthesis Mirror Optical Center True or False? In Eercises 59, determine whether the statement is true or false. Justif our answer. 59. 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. 0. In the standard form of the equation of a hperbola, the trivial solution of two intersecting lines occurs when b 0.. If D 0 and E 0, then the graph of D E 0 is a hperbola.. If the asmptotes of the hperbola a b, where a, b > 0, intersect at right angles, then a b.. Think About It Consider a hperbola centered at the origin with a horizontal transverse ais. Use the definition of a hperbola to derive its standard form.. Writing Eplain how the central rectangle of a hperbola can be used to sketch its asmptotes. 5. Use the figure to show that a.. Think About It Find the equation of the hperbola for an point on which, the difference between its distances from the points, and 0, is. 7. Proof Show that c a b for the equation of the hperbola a b where the distance from the center of the hperbola to a focus is c. 8. Proof Prove that the graph of the equation A C D E F 0 is one of the following (ecept in degenerate cases). Conic Condition (a) Circle A C (b) Parabola A 0 or C 0 (but not both) (c) Ellipse AC > 0 (d) Hperbola AC < 0 Skills Review ( c, 0) (c, 0) ( a, 0) d d (a, 0) (, ) In Eercises 9 7, perform the indicated operation. 9. Subtract: 70. Multipl: 7. Divide: 7. Epand: In Eercises 7 78, factor the polnomial completel d d 0, 0

32 90 Chapter 9 Topics in Analtic Geometr 9. Rotation and Sstems of Quadratic Equations Rotation In the preceding section, ou learned that the equation of a conic with aes parallel to the coordinate aes has a standard form that can be written in the general form A C D E F 0. Horizontal or vertical aes 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 0 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 9.9. What ou should learn Rotate the coordinate aes to eliminate the -term in equations of conics. Use the discriminant to classif conics. Solve sstems of quadratic equations. Wh ou should learn it As illustrated in Eercises on page 97, rotation of the coordinate aes can help ou identif the graph of a general second-degree equation. θ Figure 9.9 After the rotation, the equation of the conic in the new form A C D E F 0. -plane will have the 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 (See the proof on page 78.) The general second-degree equation A B C D E F 0 can be rewritten as A C D E F 0 b rotating the coordinate aes through an angle, where The coefficients of the new equation are obtained b making the substitutions cos sin and sin cos. cot A C B.

33 Section 9. Rotation and Sstems of Quadratic Equations 9 Eample Rotation of Aes for a Hperbola Rotate the aes to eliminate the -term in the equation 0. Then write the equation in standard form and sketch its graph. Solution Because A 0, B, and C 0, ou have which implies that and cot A C B The equation in the -sstem is obtained b substituting these epressions into the equation cos sin sin cos 0 0 Write in standard form. In the -sstem, this is a hperbola centered at the origin with vertices at ±, 0, as shown in Figure 9.0. To find the coordinates of the vertices in the -sstem, substitute the coordinates ±, 0 into 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. ±, ( ) ( ) = ( ( = 0 Vertices: In -sstem:, 0,, 0 In -sstem:,,, Figure 9.0 STUDY TIP 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 made a mistake. cos sin sin cos ( (

34 9 Chapter 9 Topics in Analtic Geometr Eample Rotation of Aes for an Ellipse Rotate the aes to eliminate the -term in the equation 7 0. Then write the equation in standard form and sketch its graph. Prerequisite Skills To review conics, see Sections Solution Because A 7, B, and C, ou have which implies that The equation in the -sstem is obtained b making the substitutions and into 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 ±, 0 in the -sstem, as shown in Figure 9.. cot A C B cos sin sin cos Now tr Eercise. ( ) ( ) + = 7 + = 0 Vertices: In -sstem: ±, 0, 0, ± In -sstem: Figure ,,,,,,,

35 Section 9. Rotation and Sstems of Quadratic Equations 9 Eample Rotation of Aes for a Parabola Rotate the aes to eliminate the -term in the equation Then write the equation in standard form and sketch its graph. 5 Solution Because A, B, and C, ou have cot A C B Using the identit cot cot cot produces from which ou obtain the equation. cot cot cot cot cot cot cot 0 cot cot 0. Considering 0 < <, ou have cot. So, θ Figure = 0 θ. cot From the triangle in Figure 9., ou obtain sin 5 and cos 5. So, ou use the substitutions cos sin sin cos Substituting these epressions into the original equation, ou have which simplifies as follows Group terms. Write in completed square form. Write in standard form. The graph of this equation is a parabola with verte at 5, in the -plane. Its ais is parallel to the -ais in the -sstem, as shown in Figure Now tr Eercise. ( + ) = ( ) 5) Verte: In -sstem: 5, 0 In -sstem: Figure 9. ) 55, 55

36 9 Chapter 9 Topics in Analtic Geometr 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 that is, Such quantities are invariant under rotation. The net theorem lists some other rotation invariants. F F. Rotation Invariants The rotation of the coordinate aes through an angle that transforms the equation A B C D E F 0 into the form A C D E F 0 has the following rotation invariants.. F F. A C A C. B AC B AC You can use the results of this theorem to classif the graph of a seconddegree equation with an -term in much the same wa that ou classif the graph of 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 0. Now, from the classification procedure given in Section 9., ou know that the sign of determines the tpe of graph for the equation AC B 0, A C D E F 0. Consequentl, the sign of B AC will determine the tpe of graph for the original equation, as shown in the following classification. Classification of Conics b the Discriminant The graph of the equation A B C D E F 0 is, ecept in degenerate cases, determined b its discriminant as follows.. Ellipse or circle: B AC < 0. Parabola: B AC 0. Hperbola: B AC > 0 For eample, in the general equation , ou have A, B 7, and C 5. So, the discriminant is B AC Because < 0, the graph of the equation is an ellipse or a circle.

37 Eample Rotations 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. 0 b. 9 0 c Section 9. Rotation and Sstems of Quadratic Equations 95 Solution a. Because B AC 9 < 0, the graph is a circle or an ellipse. Solve for as follows. 0 Write original equation. 0 Quadratic form a b c 0 ± ± 7 5 Graph both of the equations to obtain the ellipse shown in Figure 9.. Figure 9. 7 Top half of ellipse 7 Bottom half of ellipse b. Because B AC 0, the graph is a parabola. 9 0 Write original equation. 9 0 Quadratic form a b c 0 ± 9 9 ± 9 Graph both of the equations to obtain the parabola shown in Figure Figure 9.5 c. Because B AC 8 > 0, the graph is a hperbola Write original equation Quadratic form a b c ± ± 7 0 Graph both of the equations to obtain the hperbola shown in Figure 9.. Figure 9. Now tr Eercise 7.

38 9 Chapter 9 Topics in Analtic Geometr Sstems of Quadratic Equations To find the points of intersection of two conics, ou can use elimination or substitution, as demonstrated in Eamples 5 and. Eample 5 Solving a Quadratic Sstem b Elimination Solve the sstem of quadratic equations Equation Equation TECHNOLOGY SUPPORT For instructions on how to use the intersect feature, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Algebraic Solution You can eliminate the -term b adding the two equations. The resulting equation can then be solved for There are two real solutions: and 5. The corresponding -values are 0 and ±. So, the solutions of the sstem are, 0, 5,, and 5,. Graphical Solution Begin b solving each equation for as follows. ± 9 ± 9 Use a graphing utilit to graph all four equations 9, and 9, 9, 9 in the same viewing window. Use the intersect feature of the graphing utilit to approimate the points of intersection to be, 0, 5,, and 5,, as shown in Figure = = Now tr Eercise. = 9 Figure = + 9 Eample Solving a Quadratic Sstem b Substitution Solve the sstem of quadratic equations Equation Equation Solution Because Equation has no -term, solve the equation for tobtain. Net, substitute this into Equation and solve for In factored form, ou can see that the equation has two real solutions: 0 and. The corresponding values of are and 0. This implies that the solutions of the sstem of equations are 0, and, 0, as shown in Figure 9.8. Now tr Eercise = 0 (0,) 5 (, 0) + = 0 Figure 9.8

39 Section 9. Rotation and Sstems of Quadratic Equations Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. 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 0. In Eercises and, 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. 90, 0,.. 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 5 0, 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 ,, In Eercises, match the graph with its equation. [The graphs are labeled (a), (b), (c), (d), (e), and (f ).] (a) (c) (e) (b) (d) (f) In Eercises 7, (a) use the discriminant to classif the graph of the equation, (b) use the Quadratic Formula to solve for, and (c) use a graphing utilit to graph the equation

40 98 Chapter 9 Topics in Analtic Geometr In Eercises 5 8, sketch (if possible) the graph of the degenerate conic In Eercises 9, solve the sstem of quadratic equations algebraicall b the method of elimination. Then verif our results b using a graphing utilit to graph the equations and find an points of intersection of the graphs In Eercises 7 5, solve the sstem of quadratic equations algebraicall b the method of substitution. Then verif our results b using a graphing utilit to graph the equations and find an points of intersection of the graphs Snthesis True or False? In Eercises 5 and 5, determine whether the statement is true or false. Justif our answer. 5. The graph of k 0 0, where k is an constant less than, is a hperbola. 5. After using a rotation of aes to eliminate the -term from an equation of the form the coefficients of the - and -terms remain A and C, respectivel. Skills Review In Eercises 55 58, sketch the graph of the rational function. Identif all intercepts and asmptotes. 55. g 5. f 57. ht t 58. t In Eercises 59, if possible, find (a) AB, (b) BA, and (c) A A B C D E F 0 A B A 5 0, A 5, B A 0 5, B , In Eercises 70, graph the function. f f g g 7. ht t 8. ht t 9. ft t ft t In Eercises 7 7, find the area of the triangle. 7. C 0, a 8, b 7. B 70, a 5, c 7. a, b 8, c 0 7. a, b 5, c B gs s 0 5

41 Section 9.5 Parametric Equations Parametric Equations Plane Curves Up to this point, ou have been representing a graph b a single equation involving two variables such as 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 of 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 Rectangular equation 7 as shown in Figure 9.9. However, this equation does not tell the whole stor. Although it does tell ou where the object has been, it doesn t 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 Parametric equation for t t. Parametric equation for From this set of equations ou can determine that at time t 0, the object is at the point 0, 0. Similarl, at time t, the object is at the point,, and so on. What ou should learn Evaluate sets of parametric equations for given values of the parameter. Graph 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 57 on page 70, a set of parametric equations is used to model the path of a football. Rectangular equation: = + 7 Parametric equations: = t = t + t 8 9 (0, 0) t = t = t = (, 8) (7, 0) Elsa/Gett Images Curvilinear motion: two variables for position, one variable for time Figure 9.9 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 a Plane Curve If f and g are continuous functions of t on an interval l, the set of ordered pairs f t, gt is a plane curve C. The equations given b f t and gt are parametric equations for C, and t is the parameter.

42 700 Chapter 9 Topics in Analtic Geometr Sketching a Plane Curve One wa to sketch a curve represented b a pair of parametric equations is to plot points in the -plane. Each set of coordinates, is determined from a value chosen for the parameter t. B plotting the resulting points in the order of increasing values of t, ou trace the curve in a specific direction. This is called the orientation of the curve. Eample Sketching a Plane Curve Sketch the curve given b the parametric equations t and t, t. Describe the orientation of the curve. Solution Using values of t in the interval, the parametric equations ield the points shown in the table. t , 0 B plotting these points in the order of increasing t, ou obtain the curve shown in Figure 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 0, and then move along the curve to the point 5,. Now tr Eercises 7(a) and (b). Figure 9.50 Note that the graph shown in Figure 9.50 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. Two different sets of parametric equations can 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 9.50 and 9.5, 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. Figure 9.5 TECHNOLOGY TIP Most graphing utilities have a parametric mode. So, another wa to displa a curve represented b a pair of parametric equations is to use a graphing utilit, as shown in Eample. For instructions on how to use the parametric mode, see Appendi A; for specifice kestrokes, go to this tetbook s Online Stud Center.

43 Section 9.5 Parametric Equations 70 Eample Using a Graphing Utilit in Parametric Mode Use a graphing utilit to graph the curves represented b the parametric equations. Using the graph and the Vertical Line Test, for which curve is a function of? a. t, t b. t, t c. t, t Solution Begin b setting the graphing utilit to parametric mode. When choosing a viewing window, ou must set not onl minimum and maimum values of and, but also minimum and maimum values of t. a. Enter the parametric equations for and, as shown in Figure 9.5. Use the viewing window shown in Figure 9.5. The curve is shown in Figure 9.5. From the graph, ou can see that is not a function of. Figure 9.5 Figure 9.5 Figure 9.5 b. Enter the parametric equations for and, as shown in Figure Use the viewing window shown in Figure 9.5. The curve is shown in Figure From the graph, ou can see that isa function of. Prerequisite Skills See Section. to review the Vertical Line Test. Eploration Use a graphing utilit set in parametric mode to graph the curve t and t Set the viewing window so that and. Now, graph the curve with various settings for t. Use the following. a. 0 t b. t 0 c. t Compare the curves given b the different t settings. Repeat this eperiment using t. How does this change the results? Figure 9.55 Figure 9.5 Figure 9.57 c. Enter the parametric equations for and, as shown in Figure Use the viewing window shown in Figure The curve is shown in Figure 9.0. From the graph, ou can see that is not a function of. Figure 9.58 Figure 9.59 Figure 9.0 Now tr Eercise 7(c). TECHNOLOGY TIP Notice in Eample that in order to set the viewing windows of parametric graphs, ou have to scroll down to enter the Yma and Yscl values.

44 70 Chapter 9 Topics in Analtic Geometr Eliminating the Parameter Man curves that are represented b sets of parametric equations have graphs that can also be represented b rectangular equations (in and ). The process of finding the rectangular equation is called eliminating the parameter. Parametric equations Solve for t in one equation. Substitute in second equation. Rectangular equation t t t Now ou can recognize that the equation represents a parabola with a horizontal ais and verte at, 0. 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. This situation is demonstrated in Eample. Eample Eliminating the Parameter Identif the curve represented b the equations t and Solution Solving for t in the equation for produces t or which implies that t. Substituting in the equation for, ou obtain the rectangular equation t t t t. t. From the rectangular equation, ou can recognize that the curve is a parabola that opens downward and has its verte at 0,, as shown in Figure 9.. The rectangular equation is defined for all values of. The parametric equation for, however, is defined onl when t >. From the graph of the parametric equations, ou can see that is alwas positive, as shown in Figure 9.. So, ou should restrict the domain of to positive values, as shown in Figure 9.. Now tr Eercise 7(d). STUDY TIP 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 determine the position, direction, and speed at a given time. Figure 9. = t = t = 0 t = 0.75 Figure 9. Parametric equations: =, = t t + t + Rectangular equation: =, > 0 Figure 9.

45 Section 9.5 Parametric Equations 70 Eample Eliminating the Parameter Sketch the curve represented b cos and sin, 0, b eliminating the parameter. Solution Begin b solving for cos and sin in the equations. cos and sin Solve for cos and sin. Use the identit sin cos to form an equation involving onl and. cos sin 9 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 0, 0, with vertices 0, and 0,, and minor ais of length b, as shown in Figure 9.. Note that the elliptic curve is traced out counterclockwise. Now tr Eercise. Eploration In Eample, ou make use of the trigonometric identit sin cos to sketch an ellipse. Which trigonometric identit would ou use to obtain the graph of a hperbola? Sketch the curve represented b sec and tan, 0, b eliminating the parameter. Figure 9. 5 = cos θ = sin θ θ = 7 θ = 0 θ = 8 θ = 5 Finding Parametric Equations for a Graph How can ou determine 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. This is further demonstrated in Eample 5. Eample 5 Finding Parametric Equations for a Given Graph Find a set of parametric equations to represent the graph of using the parameters (a) t and (b) t. Solution a. Letting t, ou obtain the following parametric equations. t t Parametric equation for Parametric equation for The graph of these equations is shown in Figure 9.5. b. Letting t, ou obtain the following parametric equations. t t t t Parametric equation for Parametric equation for The graph of these equations is shown in Figure 9.. Note that the graphs in Figures 9.5 and 9. have opposite orientations. Now tr Eercise 5. t = 0 t = t = t = Figure 9.5 t = = t = t t = t = t = 0 t = Figure 9. t = = t = t t

46 70 Chapter 9 Topics in Analtic Geometr 9.5 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. If f and g are continuous functions of t on an interval I, the set of ordered pairs ft, gt is a C. The equations given b ft and gt are for C, and t is the.. 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 rectangular form is called the. In Eercises, match the set of parametric equations with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (c) (e) (b) (d) (f). t, t.. t, t. 5. ln t, t Consider the parametric equations t and t. (a) Create a table of - and -values using t 0,,,, and. (b) Plot the points, generated in part (a) and sketch a graph of the parametric equations. (c) Use a graphing utilit to graph the curve represented b the parametric equations. (d) Find the rectangular equation b eliminating the parameter. Sketch its graph. How does the graph differ from those in parts (b) and (c)? t, t t t, t, e t 8. Consider the parametric equations cos and sin. (a) Create a table of - and -values using, 0,, and. (b) Plot the points, generated in part (a) and sketch a graph of the parametric equations. (c) Use a graphing utilit to graph the curve represented b the parametric equations. (d) Find the rectangular equation b eliminating the parameter. Sketch its graph. How does the graph differ from those in parts (b) and (c)? Librar of Parent Functions In Eercises 9 and 0, determine the plane curve whose graph is shown. 9. (a) (b) t (c) 0. (a) (d) t (b) cos (c) t t t t t t cos sin sin cos sin (d) cos sin 8 8 8,

47 Section 9.5 Parametric Equations 705 In Eercises, sketch the curve represented b the parametric equations (indicate the orientation of the curve). Use a graphing utilit to confirm our result. Then eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessar.. t, t.. t, t. 5. t, t. 7. t, t t, t 0.. cos, sin.. e t, e t. 5. t, ln t. In Eercises 7, use a graphing utilit to graph the curve represented b the parametric equations. 7. cos 8. cos sin tan lnt In Eercises and, determine how the plane curves differ from each other. t cos (c) e t (d) e t (c) e t t t t In Eercises 5 8, eliminate the parameter and obtain the standard form of the rectangular equation. 5. Line through, and, : t t. Circle: 7. Ellipse: 8. Hperbola: (d) t, t 9. sec 0. sec. (a) t (b) cos e t. (a) t (b) t t t t h r cos, k r sin h a cos, k b sin t, t t, t t, t t, t cos, sin e t, e t ln t, t sin tan. t e t 0.t h a sec, k b tan In Eercises 9, use the results of Eercises 7 0 to find a set of parametric equations for the line or conic. 9. Line: passes through, and, 0. Circle: center:, 5; radius:. Ellipse: vertices: ±5, 0; foci: ±, 0. Hperbola: vertices: 0, ±; foci: 0, ± In Eercises 8, find two different sets of parametric equations for the given rectangular equation. (There are man correct answers.) In Eercises 9 and 50, use a graphing utilit to graph the curve represented b the parametric equations. 9. Witch of Agnesi: cot, sin 50. Folium of Descartes: t t t, t In Eercises 5 5, match the parametric equations with the correct graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) 5 5 (b) (d) 5. Lissajous curve: cos, sin 5. Evolute of ellipse: cos, sin 5. Involute of circle: cos sin sin cos 5. Serpentine curve: cot, sin cos

48 70 Chapter 9 Topics in Analtic Geometr Projectile Motion In Eercises 55 58, consider a projectile launched at a height of h feet above the ground at an angle of with the horizontal. The initial velocit is v 0 feet per second and the path of the projectile is modeled b the parametric equations v and h v 0 sin t t 0 cos t. 55. The center field fence in a ballpark is 0 feet high and 00 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 00 miles per hour (see figure). (a) Write a set of parametric equations for the path of the baseball. (b) Use a graphing utilit to graph the path of the baseball for Is the hit a home run? (c) Use a graphing utilit to graph the path of the baseball for Is the hit a home run? (d) Find the minimum angle required for the hit to be a home run. 5. The right field fence in a ballpark is 0 feet high and feet from home plate. A baseball is hit at a point.5 feet above the ground. It leaves the bat at an angle of with the horizontal at a speed of 05 feet per second. (a) Write a set of parametric equations for the path of the baseball. (b) Use a graphing utilit to graph the path of the baseball and approimate its maimum height. (c) Use a graphing utilit to find the horizontal distance that the baseball travels. Is the hit a home run? (d) Eplain how ou could find the result in part (c) algebraicall. 57. The quarterback of a football team releases a pass at a height of 7 feet above the plaing field, and the football is caught b a receiver at a height of feet, 0 ards directl downfield. The pass is released at an angle of 5 with the horizontal. (a) Write a set of parametric equations for the path of the football. (b) Find the speed of the football when it is released. (c) Use a graphing utilit to graph the path of the football and approimate its maimum height. (d) Find the time the receiver has to position himself after the quarterback releases the football. 5.. ft θ 00 ft 0 ft Not drawn to scale To begin a football game, a kicker kicks off from his team s 5-ard line. The football is kicked at an angle of 50 with the horizontal at an initial velocit of 85 feet per second. (a) Write a set of parametric equations for the path of the kick. (b) Use a graphing utilit to graph the path of the kick and approimate its maimum height. (c) Use a graphing utilit to find the horizontal distance that the kick travels. (d) Eplain how ou could find the result in part (c) algebraicall. Snthesis True or False? In Eercises 59, determine whether the statement is true or false. Justif our answer. 59. The two sets of parametric equations t, t and t, 9t correspond to the same rectangular equation. 0. Because the graph of the parametric equations t, t and t, t both represent the line, the are the same plane curve.. If is a function of t and is a function of t, then must be a function of.. The parametric equations at h and bt k, where a 0 and b 0, represent a circle centered at h, k, if a b.. As increases, the ellipse given b the parametric equations cos and sin is traced out counterclockwise. Find a parametric representation for which the same ellipse is traced out clockwise.. Think About It The graph of the parametric equations t and t is shown below. Would the graph change for the equations t and t? If so, how would it change? Skills Review In Eercises 5 8, check for smmetr with respect to both aes and the origin. Then determine whether the function is even, odd, or neither. 5. f. f 7. e 8. 5

49 Section 9. Polar Coordinates Polar Coordinates Introduction So far, ou have been representing graphs of equations as collections of points, in the rectangular coordinate sstem, where and represent the directed distances from the coordinate aes to the point,. In this section, ou will stud a second coordinate 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 9.7. 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 the polar ais to segment OP O r = directed distance θ Figure 9.7 P = ( r, θ) = directed angle Polar ais What ou should learn Plot points and find multiple representations of points in 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 on page 7, ou see that a polar coordinate can be written in more than one wa. Eample Plotting Points in 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 9.8. b. The point r,, lies three units from the pole on the terminal side of the angle as shown in Figure 9.9. c. The point r,, coincides with the point,, as shown in Figure 9.70.,, θ = (, ) 0 Figure 9.8 Figure 9.9 Now tr Eercise 5. 0 θ = (, ) 0 θ = Figure 9.70 (, )

50 708 Chapter 9 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, ± n where n is an integer. Moreover, the pole is represented b 0,, where is an angle. Eample Multiple Representations of Points Plot the point, and find three additional polar representations of this point, using < <. Solution The point is shown in Figure 9.7. Three other representations are as follows., 5,, 7,,, Now tr Eercise 7. Add to. Replace r b r; subtract from. Replace r b r; add to. 0 (, ) θ = ( ) ( ) ( ) ( ), =, 5 =, 7 =, =... Figure 9.7 ( r, θ) (, ) Coordinate Conversion r 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 9.7. Because, lies on a circle of radius r, it follows that r. Moreover, for r > 0, the definitions of the trigonometric functions impl that Pole θ (Origin) Figure 9.7 Polar ais ( -ais) tan, cos r, and sin r. You can show that the same relationships hold for r < 0. Coordinate Conversion The polar coordinates r, are related to the rectangular coordinates, as follows. r cos and tan r sin r

51 Section 9. Polar Coordinates 709 Eample Polar-to-Rectangular Conversion Convert each point to rectangular coordinates. a., b., Solution a. For the point r,,, ou have the following. r cos cos r sin sin 0 The rectangular coordinates are,, 0. (See Figure 9.7.) b. For the point r,,, ou have the following. r cos cos r sin sin The rectangular coordinates are,,. (See Figure 9.7.) Eample Now tr Eercise. Rectangular-to-Polar Conversion Convert each point to polar coordinates. a., b. 0, Eploration Set our graphing utilit to polar mode. Then graph the equation r. Use a viewing window in which 0,, and. You should obtain a circle of radius. a. Use the trace feature to cursor around the circle. Can ou locate the point, 5? b. Can ou locate other representations of the point, 5? If so, eplain how ou did it. ( r, θ) = (, ) (, ) = (, 0) ( ) ( r, θ) =, (, ) = (, ) Solution 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 9.7. b. Because the point, 0, lies on the positive -ais, choose. and r. This implies that one set of polar coordinates is r,,, as shown in Figure Now tr Eercise 9. Figure 9.7 (, ) = (, ) ( r, θ) =, Figure 9.7 ( ) (, ) = (0, ) Figure 9.75 ( ) ( r, θ) =, 0 0

52 70 Chapter 9 Topics in Analtic Geometr Equation Conversion B comparing Eamples and, ou 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. r sin r cos r sec tan 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. Solution 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 9.7. 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 -ais, as shown in Figure To convert to rectangular form, ou 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 ou convert to rectangular form b using the relationship r cos. r sec a. r b. c. r sec r tan r cos 0 Figure Figure Polar equation Rectangular equation Now ou can see that the graph is a vertical line, as shown in Figure Now tr Eercise 8. Figure 9.78

53 Section 9. Polar Coordinates 7 9. Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The origin of the polar coordinate sstem is called the.. For the point r,, r is the from O to P and is the counterclockwise from the polar ais to segment OP.. To graph the point r,, ou use the coordinate sstem. In Eercises, a point in polar coordinates is given. Find the corresponding rectangular coordinates for the point..,. 5.,. In Eercises 5, plot the point given in polar coordinates and find three additional polar representations of the point, using < <. 5 5.,., 7., , 0..,. 0 5 ( ) ( r, θ) =, ( ) ( r, θ) =, 0,, 7 5,, 0, ( ) ( r, θ) =, 0 0 ( ) ( r, θ) =, In Eercises 0, plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point.. 7,., 5.,., , , 9.,. 0.,.57 In Eercises 8, use a graphing utilit to find the rectangular coordinates of the point given in polar coordinates. Round our results to two decimal places..,., ,.. 8.5, , , , ,. In Eercises 9, plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for 0 } <. 9. 7, , 5.,.,.,., 5., 9. 5, In Eercises 7, use a graphing utilit to find one set of polar coordinates for the point given in rectangular coordinates. (There are man correct answers.) 7., 8. 5, 9., 0.,. 5. 7,,

54 7 Chapter 9 Topics in Analtic Geometr In Eercises 0, convert the rectangular equation to polar form. Assume a > a a a In Eercises 80, convert the polar equation to rectangular form.. r sin. r cos r 70. r 0 7. r csc 7. r sec 7. r cos 7. r sin 75. r sin 7. r cos 77. r 78. r cos sin 79. r 80. r sin cos sin In Eercises 8 8, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. 8. r 7 8. r r sec 8. r csc Snthesis True or False? In Eercises 87 and 88, determine whether the statement is true or false. Justif our answer. 87. If r, and r, represent the same point in the polar coordinate sstem, then r r. 88. If r, and r, represent the same point in the polar coordinate sstem, then n for some integer n Think About It (a) Show that the distance between the points r, and r, is (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 90. Is the simplification what ou epected? Eplain. (d) Choose two points in 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. 90. Writing Write a short paragraph eplaining the differences between the rectangular coordinate sstem and the polar coordinate sstem. Skills Review r r r r cos. In Eercises 9 9, use the Law of Sines or the Law of Cosines to solve the triangle. 9. a, b 9, c 5 9. A, a 0, b 9. A 5, C 8, c 9. B 7, a, c C 5, a 8, b 9. B, b 5, c In Eercises 97 0, use an method to solve the sstem of equations u 7v 9w 5 u v w 7 a b c 0 a b c 0 a b 9c 0 8u v w z z 5 z In Eercises 0 0, use a determinant to determine whether the points are collinear. 0.,,, 7,, 0.,, 0,,,5 05.,,,,.5,.5 0.., 5, 0.5, 0,.5,

55 Section 9.7 Graphs of Polar Equations Graphs of Polar Equations Introduction In previous chapters ou sketched graphs in rectangular coordinate sstems. You began with the basic point-plotting method. Then ou used sketching aids such as a graphing utilit, smmetr, intercepts, asmptotes, periods, and shifts to further investigate the nature of the graph. This section approaches curve sketching in the polar coordinate sstem similarl. Eample Graphing a Polar Equation b Point Plotting Sketch the graph of the polar equation r sin b hand. Solution The sine function is periodic, so ou can get a full range of r-values b considering values of in the interval 0, as shown in the table. 0 r B plotting these points as shown in Figure 9.79, it appears that the graph is a circle of radius whose center is the point, 0,. 5 7 What ou should learn Graph polar equations b point plotting. Use smmetr as a sketching aid. Use zeros and maimum r-values as sketching aids. Recognize special polar graphs. Wh ou should learn it Several common figures, such as the circle in Eercise on page 70, are easier to graph in the polar coordinate sstem than in the rectangular coordinate sstem. Prerequisite Skills If ou have trouble finding the sines of the angles in Eample, review Trigonometric Functions of An Angle in Section.. Figure 9.79 Now tr Eercise 5. You can confirm the graph found in Eample in three was.. Convert to Rectangular Form Multipl each side of the polar equation b r and convert the result to rectangular form.. Use a Polar Coordinate Mode Set our graphing utilit to polar mode and graph the polar equation. (Use 0,, and.). Use a Parametric Mode Set our graphing utilit to parametric mode and graph sin t cos t and sin t sin t.

56 7 Chapter 9 Topics in Analtic Geometr Most graphing utilities have a polar graphing mode. If ours doesn t, ou can rewrite the polar equation r f in parametric form, using t as a parameter, as follows. f t cos t and f t sin t Smmetr In Figure 9.79, note that as increases from 0 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. The three important tpes of smmetr to consider in polar curve sketching are shown in Figure (, r θ) θ ( r, θ) ( r, θ) θ 0 ( r, θ) θ θ 0 + θ ( r, θ) θ 0 (, r θ) ( r, θ) ( r, θ) (, r + θ) Smmetr with Respect to the Line Figure 9.80 Smmetr with Respect to the Polar Ais Smmetr with Respect to the Pole Testing 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,. You can determine the smmetr of the graph of r sin (see Eample ) as follows.. Replace r, b r, : r sin r sin sin. Replace r, b r, : r sin sin. Replace r, b r, : r sin r sin So, the graph of r sin is smmetric with respect to the line. STUDY TIP Recall from Section. that the sine function is odd. That is, sin sin.

57 Section 9.7 Graphs of Polar Equations 75 Eample Using Smmetr to Sketch a Polar Graph Use smmetr to sketch the graph of r cos b hand. Solution Replacing r, b r, produces r cos cos. cosu cos u So, b using the even trigonometric identit, 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 9.8. This graph is called a limaçon. Use a graphing utilit to confirm this graph. The three tests for smmetr in polar coordinates on page 7 are sufficient to guarantee smmetr, but the are not necessar. For instance, Figure 9.8 shows the graph of r Spiral of Archimedes From the figure, ou can see that the graph is smmetric with respect to the line Yet the tests on page 7 fail to indicate smmetr because neither of the following replacements ields an equivalent equation. Original Equation Replacement New Equation r r, b r, r r r, b r, r The equations discussed in Eamples and are of the form r sin fsin and 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 quick tests for smmetr... 0 Now tr Eercise 9., Quick Tests for Smmetr in Polar Coordinates. The graph of r fsin is smmetric with respect to the line.. The graph of r gcos is smmetric with respect to the polar ais. 5 r 5 Figure 9.8 TECHNOLOGY TIP The table feature of a graphing utilit is ver useful in constructing tables of values for polar equations. Set our graphing utilit to polar mode and enter the polar equation in Eample. You can verif the table of values in Eample b starting the table at and incrementing the value of b. For instructions on how to use the table feature and polar mode, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Spiral of Archimedes: r = +, 0 θ Figure θ

58 7 Chapter 9 Topics in Analtic Geometr Zeros and Maimum r-values Two additional aids to sketching graphs of polar equations involve knowing the -values for which r is maimum and knowing the -values for which r 0. In Eample, the maimum value of r for r sin is r, and this occurs when (see Figure 9.79). Moreover, r 0 when 0. Eample Finding Maimum r-values of a Polar Graph Find the maimum value of r for the graph of r cos. Graphical Solution Because the polar equation is of the form r cos gcos ou know the graph is smmetric with respect to the polar ais. You can confirm this b graphing the polar equation. Set our graphing utilit to polar mode and enter the equation, as shown in Figure 9.8. (In the graph, varies from 0 to. ) To find the maimum r-value for the graph, use our graphing utilit s trace feature and ou should find that the graph has a maimum r-value of, as shown in Figure 9.8. This value of r occurs when In the graph, note that the point, is farthest from the pole.. Numerical Solution To approimate the maimum value of r for the graph of r cos, use the table feature of a graphing utilit to create a table that begins at and increments b, as shown in Figure From the table, the maimum value of r appears to be when If a second table that begins at and increments b is created, as shown in Figure 9.8, the maimum value of r still appears to be when Limaçon: r = cos θ Figure 9.85 Figure 9.8 Figure 9.8 Note how the negative r-values determine the inner loop of the graph in Figure 9.8. This tpe of graph is a limaçon. Now tr Eercise 9. Figure 9.8 Eploration The graph of the polar equation r e cos cos sin 5 is called the butterfl curve, as shown in Figure a. The graph in Figure 9.87 was produced using 0. Does this show the entire graph? Eplain our reasoning. b. Use the trace feature of our graphing utilit to approimate the maimum r-value of the graph. Does this value change if ou use 0 instead of 0? Eplain. Figure 9.87

59 Some curves reach their zeros and maimum r-values at more than one point, as shown in Eample. Section 9.7 Graphs of Polar Equations 77 Eample Analzing a Polar Graph Analze the graph of r cos. Solution Smmetr: Maimum value of Zeros of r: r : With respect to the polar ais r or when r 0 when or 0,,, 0,,,,,, 5, r 0 0 B plotting these points and using the specified smmetr, zeros, and maimum values, ou can obtain the graph shown in Figure This graph is called a rose curve, and each loop on the graph is called a petal. Note how the entire curve is generated as increases from 0 to Eploration Notice that the rose curve in Eample has three petals. How man petals do the rose curves r cos and r sin have? Determine the numbers of petals for the curves r cos n and r sin n, where n is a positive integer Figure 9.88 Now tr Eercise.

60 78 Chapter 9 Topics in Analtic Geometr 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. 0 0 a a < a a b < b b < b Limaçon with Cardioid Dimpled Conve inner loop (heart-shaped) limaçon limaçon 0 0 Limaçons r a ± b cos r a ± b sin a > 0, b > 0 n = 0 a n = 0 a a 0 n = 5 a 0 n = Rose Curves n petals if n is odd n petals if n is even 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 0 a 0 a 0 0 Circles and Lemniscates a a r a cos r a sin r a sin r a cos Circle Circle Lemniscate Lemniscate

61 Section 9.7 Graphs of Polar Equations 79 Eample 5 Analzing a Rose Curve Analze the graph of r cos. Solution Tpe of curve: Rose curve with n petals Smmetr: With respect to the polar ais, the line and the pole Maimum value of Zeros of r: when r 0 when Using a graphing utilit, enter the equation, as shown in Figure 9.89 (with 0 ). You should obtain the graph shown in Figure r : r 0,,,, r = cos θ (, ) ( ), (, ) (, 0), STUDY TIP The quick tests for smmetr presented on page 75 are especiall useful when graphing rose curves. Because rose curves have the form r fsin or the form r gcos, ou know that a rose curve will be either smmetric with respect to the line or smmetric with respect to the polar ais. Figure 9.89 Figure 9.90 Now tr Eercise 7. Eample Analzing a Lemniscate Analze the graph of r 9 sin. Solution Tpe of curve: Lemniscate Smmetr: Maimum value of Zeros of r: r : With respect to the pole r when r 0 when 0, Using a graphing utilit, enter the equation, as shown in Figure 9.9 (with 0 ). You should obtain the graph shown in Figure 9.9. (, ) r = 9 sin θ (, ) Figure 9.9 Figure 9.9 Now tr Eercise 5.

62 70 Chapter 9 Topics in Analtic Geometr 9.7 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The graph of r fsin 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. In Eercises, identif the tpe of polar graph... r = cos θ r = 9 cos θ r = sin θ Librar of Parent Functions In Eercises 7 0, determine the equation of the polar curve whose graph is shown r = 5 5 sin θ 9 r = + cos θ 0 0 r = cos θ 9 7 (a) r sin (a) r cos (b) r sin (b) r sin (c) r cos (c) r cos (d) r cos (d) r sin (a) r cos (b) r cos (c) r cos (d) r cos (a) r sin (b) (c) (d) r sin In Eercises 8, test for smmetr with respect to the polar ais, and the pole.. r cos. r cos. r. r sin cos 5. r sin. r csc cos 7. r sin 8. r 5 cos /, 0 r cos r sin r In Eercises 9, find the maimum value of and an zeros of r. Verif our answers numericall. 9. r 0 0 sin 0. r cos. r cos. r sin

63 Section 9.7 Graphs of Polar Equations 7 In Eercises, sketch the graph of the polar equation. Use a graphing utilit to verif our graph.. r r sin. r cos 7. r cos 8. r sin 9. r cos 0. r sin. r 5 sin. r cos. r 5 cos. r sin 5 5. r 7 sin. r cos 5 In Eercises 7 5, use a graphing utilit to graph the polar equation. Describe our viewing window. 7. r 8 cos 8. r cos 9. r 5 sin 0. r sin. r cos. r sin. r. r sin cos 5. r cos. r 9 sin 7. r sin cos 8. r cos 9. r csc 50. r sec 5. r e 5. r e In Eercises 5 58, use a graphing utilit to graph the polar equation. Find an interval for for which the graph is traced onl once. 57. r 58. r sin In Eercises 59, 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 59. Conchoid 0. Conchoid. Hperbolic spiral. Strophoid r sec r csc r / r cos sec Snthesis sin cos 5. r cos 5. r sin r cos 5. r sin True or False? In Eercises and, determine whether the statement is true or false. Justif our answer.. The graph of r 0 sin 5 is a rose curve with five petals. 5. A rose curve will alwas have smmetr with respect to the line 5. Writing Use a graphing utilit to graph the polar equation r cos 5 n cos, 0 for the integers n 5 to n 5. As ou graph these equations, ou should see the graph change shape from a heart to a bell. Write a short paragraph eplaining which values of n produce the heart-shaped curves and which values of n produce the bell-shaped curves.. 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. (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 8 70, use the results of Eercises and Write an equation for the limaçon r sin after it has been rotated through each given angle. (a) (b) (c) (d) 9. Write an equation for the rose curve r sin after it has been rotated through each given angle. (a) (b) (c) (d) 70. Sketch the graph of each equation. (a). r sin (b) r sin 7. Eploration Use a graphing utilit to graph the polar equation r k cos for k 0, k, k, and k. Identif each graph. 7. Eploration Consider the polar equation r sin k. (a) Use a graphing utilit to graph the equation for k.5. Find the interval for for which the graph is traced onl once. (b) Use a graphing utilit to graph the equation for k.5. Find the interval for for which the graph is traced onl once. (c) Is it possible to find an interval for for which the graph is traced onl once for an rational number k? Eplain. <

64 7 Chapter 9 Topics in Analtic Geometr 9.8 Polar Equations of Conics Alternative Definition of Conics In Sections 9. and 9., ou learned that the rectangular equations of ellipses and hperbolas take simple forms when the origin lies at the center. 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 a conic that uses the concept of eccentricit (a measure of the flatness of the conic). Alternative Definition of a Conic The locus of a point in the plane that moves such 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 9.9.) What ou should learn Define conics in terms of eccentricities. 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 b polar equations. For instance, in Eercise 55 on page 77, ou will use polar equations to model the orbits of Neptune and Pluto. In Figure 9.9, note that for each tpe of conic, the focus is at the pole. Q Directri P F = (0, 0) 0 Q Directri P F = (0, 0) Ellipse: 0 < e < Parabola: e Hperbola: e > PF PF PF PF > PQ < PQ PQ Figure P Directri Q Q PQ P F = (0, 0) 0 Kevin Kelle/Gett Images Prerequisite Skills To review the characteristics of conics, see Sections Polar Equations of Conics The benefit of locating a focus of a conic at the pole is that the equation of the conic becomes simpler. Polar Equations of Conics (See the proof on page 79.) The graph of a polar equation of the form ep ep. r or. r ± e cos ± e sin is a conic, where e > 0 is the eccentricit and is the distance between the focus (pole) and the directri. p