2. Polar coordinates:

Transcription

1 Section 9. Polar Coordinates Section 9. Polar Coordinates In polar coordinates ou do not have unique representation of points. The point r, can be represented b r, ± n or b r, ± n where n is an integer. The pole is represented b, where is an angle. To convert from polar coordinates to rectangular coordinates, use the following relationships. r cos r sin To convert from rectangular coordinates to polar coordinates, use the following relationships. r ± tan If is in the same quadrant as the point,, then r is positive. If is in the opposite quadrant as the point,, then r is negative. You should be able to convert rectangular equations to polar form and vice versa. Vocabular Check. pole. directed distance, directed angle. polar. Polar coordinates: cos, sin Rectangular coordinates:,. Polar coordinates:, cos, sin Rectangular coordinates:, Houghton Mifflin Compan. All rights reserved.. Polar coordinates: 5 cos 5 sin 5,. Polar coordinates: Rectangular coordinates:, cos sin, Rectangular coordinates:,

2 Chapter 9 Topics in Analtic Geometr ) 5. (, 5 (. (, ( 7. Three additional representations: 5,, 7 5,, 5,,. Three additional points: 5,, 7,,, (, (, 7 ) 9. Three additional representations: ),,,, 5 ) 5,,, Three additional representations:, 7,.,,, Three additional points: 5,,,, Three additional points: 5,, ( 5, ) 5 7 9, 5, 5, 7 5, Houghton Mifflin Compan. All rights reserved.

3 Section 9. Polar Coordinates 5.. ) ), (, ( Three additional representations:,,,,,. Polar coordinates:, cos sin Rectangular coordinates:,, is the origin. Three additional points: An angle will do since r.. Polar coordinates: Rectangular coordinates: 7, cos 7 sin 7,, 5,, 7,, Houghton Mifflin Compan. All rights reserved. 5. Polar coordinates: cos sin Rectangular coordinates:,. Polar coordinates:, Rectangular coordinates:,, cos sin,

4 Chapter 9 Topics in Analtic Geometr cos 7 sin 7 Rectangular coordinates: 7. Polar coordinates:, 7 (origin!). Polar coordinates:, (origin!) cos 5, sin 5 5 Rectangular coordinates:, 9. Polar coordinates: Rectangular coordinates:,. cos.. sin..99.,.99. Polar coordinates: cos.57. sin.57. Rectangular coordinates:,.57.,. r,, 9,.5,.9. r,, 9,.,.57. r,.5,.,.,.. r,.5,.5, 7.7,.9 5. r,.5,.5,.,.5. r, 5.,.5, 5.7, r,.,.5,.,.97. r,.,.,.9,. 9. Rectangular coordinates: r 7, tan, 7, Polar coordinates: 7,, 7, Houghton Mifflin Compan. All rights reserved.

5 Section 9. Polar Coordinates 7. Rectangular coordinates: r 5, tan undefined, Polar coordinates:, 5 5,, 5,. Rectangular coordinates:, Polar coordinates:, r, tan,, 5, 5. Rectangular coordinates: r, tan, Polar coordinates:, 5,,,. Rectangular coordinates: Polar coordinates:, r, tan, 5,,,. Rectangular coordinates:, 5.,, 9 r r 9 7. Houghton Mifflin Compan. All rights reserved. tan, Polar coordinates: 5 5,, 5, tan 9 Polar coordinates:.,.9,.,

6 Chapter 9 Topics in Analtic Geometr. Rectangular coordinates: 5, 7. r 5, tan 5,.7 Polar coordinates:,.7,,.,, r r,,.5 arctan.5., 5, r, 5.9,.7.,, r,,.,.75 9.,, r 7 r, 7,.57 arctan.57., 5, r 5 7 r, 7,.9 arctan.9 5., 7, r,., r 9 r 5. r r 7.. r sin r cos sin cos tan r cos r sec 5. r cos r sin r cos sin r sin cos r sin r csc a r cos a 7 r cos 7r sin r cos 7 sin r r a sec cos 7 sin Houghton Mifflin Compan. All rights reserved.

7 Section 9. Polar Coordinates r cos r sin r cos sin r cos sin r sin r csc r cos r sin r sec csc r sec csc csc r 9r cos r sin r 9cos sin r 9 cos r r sin r cos r cos r cos r cos r cos r cos r cos r or r cos r ±r cos a r r cos r r sin r ar cos r r cos rr sin rr a cos r cos r sin r a cos 5. a 59.. r a r sin r sin r cos r cos r sin rr a sin r a sin sin r cos r sin cos r cos cot sin csc Houghton Mifflin Compan. All rights reserved.. r sin r r sin. tan sec r cos r r cos. tan tan

8 Chapter 9 Topics in Analtic Geometr tan tan 5 tan tan 5 tan tan 7., vertical line. horizontal line,9. r r 7. r 7. r csc 7. r sec r r sin r cos 7. r cos 7. r sin sin cos r r cos r r r r r ) 75. r sin 7. r cos r sin sin r cos sin r r sin r sin r r cos r sin or r r r cos cos 7. r r r sin sin Houghton Mifflin Compan. All rights reserved.

9 Section 9. Polar Coordinates 79. r sin. r cos sin r sin ± 9 5 r r sin r r r r r. r 7. r 9 9 The graph is a circle centered at the origin with radius 7. r r Circle of radius centered at origin. tan tan The graph is the line, which makes an angle of with the positive -ais.. 7 tan tan 7 Line through origin making angle of with positive -ais Houghton Mifflin Compan. All rights reserved. 5. r sec r cos Vertical line. r csc r sin Horizontal line through, 7. True, the distances from the origin are the same.. False. For instance when r, an value of gives the same point.

10 Chapter 9 Topics in Analtic Geometr 9. (a) r,, where r cos and r sin. r,, where r cos and r sin. Then and r cos r sin r r. Thus, d r r r r cos cos r r sin sin r r r r cos. (b) If (c) If, the points are on the same line through the origin. In this case, d r r r r cos r r r r. 9, d r r, the Pthagorean Theorem. 7 (d) For instance, gives, gives d.5. (Same!),, d.5 and, 9. Answers will var.,, 9. cos A b c a bc A.7 cos B a c b ac B C A, a, b sin B b sin A a C A B.9 c a sin C sin A. B B 5 9. a sin A b sin B c sin C c sin C a c sin A sin C b c sin B sin C sin5. sin sin 9. sin c a b ab cos C 9. cos c 5.5 b sin B c sin C sin B b sin C c B 5.9 A B C 9. sin5 5.5 B 7, a, c 9 b a c ac cos B 5.5 b 9.7 sin C c sin B b A B C.9 B, b 5, c sin C c sin B b A B C.5 a b sin A sin B.9 C C Houghton Mifflin Compan. All rights reserved.

11 Section 9. Polar Coordinates 97. B Cramer s Rule, 5 7 D 5 D 7 5 D 5 D, D D D Solution:, 9. B Cramer s Rule, D 5 D D D, D 5, D D Solution: 5, Houghton Mifflin Compan. All rights reserved. 99. B Cramer s Rule, 9 D 5 9 D a 9 D b D c a D a, b D b, D D Solution:,, c D c D. B Cramer s Rule, D D u D v D w 7 7 u D u v D v D 9 9, D w D w D Solution: 95 9, 9, ,

12 Chapter 9 Topics in Analtic Geometr. B Cramer s Rule, D 5 D D 5 D z D D z D z D 5, Solution:,, D D 5, 5. Cramer s Rule does not appl because D. Use elimination to solve the sstem. 5 R R R R R R R R Let then a, a and a Solution: a, a, a.. Points:,,, 7,, 7 The points are not collinear.. Points:,,,,, 5 5 collinear 5. Points:,,,,.5, The points are collinear.. Points:..5.5., 5,.5,,.5, 5. not collinear Houghton Mifflin Compan. All rights reserved.

13 Section 9.7 Graphs of Polar Equations 5 Section 9.7 Graphs of Polar Equations When graphing polar equations:. Test for smmetr : (a) Replace r, b r, or r,. (b) Polar ais: Replace r, b r, or r,. (c) Pole: Replace r, b r, or r,. (d) r f sin is smmetric with respect to the line (e) r f cos is smmetric with respect to the polar ais. r. Find the values for which is maimum.. Find the values for which r... Know the different tpes of polar graphs. (a) Limaçons (b) Rose curves, n (c) Circles (d) Lemniscates r a ± b cos r a cos n r a cos r a cos r a ± b sin r a sin n r a sin r a sin r a You should be able to graph polar equations of the form r f with our graphing utilit. If our utilit does not have a polar mode, use f t cos t f t sin t in parametric mode. Vocabular Check.. polar ais. conve limaçon. circle 5. lemniscate. cardioid Houghton Mifflin Compan. All rights reserved.. r cos is a rose curve.. r cos is a circle.. Cardioid 7. The graph is smmetric about the line and passes through r,,. Matches (a). 5. r sin is a rose curve.. Lemniscate. Limaçon,. The graph is smmetric about the polar ais and passes through r,,. Matches (c). 9. The graph has four leaves. Matches (c).. The graph has three leaves. Matches (d).

14 Chapter 9 Topics in Analtic Geometr. r cos : r cos. r cos : r cos r cos r cos Not an equivalent equation Not an equivalent equation r cos r cos cos sin sin r cos r cos r cos Not an equivalent equation Polar ais: Not an equivalent equation r cos Polar ais: r cos r cos r cos Equivalent equation Equivalent equation Pole: r cos Pole: r cos Not an equivalent equation Not an equivalent equation r cos r cos r cos r cos Not an equivalent equation Not an equivalent equation Answer: Smmetric with respect to polar ais Answer: Smmetric with respect to polar ais. r sin : r sin Pole: r sin Polar ais: r r Equivalent equation r r Not an equivalent equation r r sin cos cos sin sin sin sin sin sin Not an equivalent equation Not an equivalent equation r r sin sin Not an equivalent equation Answer: Smmetric with respect to Houghton Mifflin Compan. All rights reserved.

15 Section 9.7 Graphs of Polar Equations 7. r cos : Polar ais: r r Not an equivalent equation r r r Not an equivalent equation r r cos cos cos cos cos sin sin cos cos cos Equivalent equation Pole: r Not an equivalent equation r r r cos cos cos cos sin sin cos Not an equivalent equation Answer: Smmetric with respect to the polar ais 5. r sin. r csc cos cot : r sin : r cot r sin r cot Equivalent equation Equivalent equation Polar ais: r sin Polar ais: r cot r sin r cot Not an equivalent equation r cot Houghton Mifflin Compan. All rights reserved. r sin r sin cos cos sin r sin Not an equivalent equation Pole: r sin Not an equivalent equation r sin r sin Not an equivalent equation Answer: Smmetric with respect to Equivalent equation Pole: r cot r cot Equivalent equation Answer: Smmetric with respect to polar ais and pole,

16 Chapter 9 Topics in Analtic Geometr 7. r sin. r 5 cos : r sin : r 5 cos r sin r 5 cos Not an equivalent equation Equivalent equation r sin Polar ais: r 5 cos r sin r 5 cos r sin Equivalent equation Not an equivalent equation Pole: r 5 cos Polar ais: r sin r 5 cos r sin Equivalent equation Not an equivalent equation r sin r sin Answer: Smmetric with respect to polar ais and pole, Not an equivalent equation Pole: r sin r sin Equivalent equation Answer: Smmetric with respect to pole 9. r sin. sin r cos cos cos sin sin or sin cos sin sin Maimum: r when r when sin sin or Not possible Maimum: r when Zero: r when, Houghton Mifflin Compan. All rights reserved.

17 Section 9.7 Graphs of Polar Equations 9. r cos cos. cos cos ± Maimum:,,, r when r when cos,,, 5,, r sin r sin Maimum: r when,, 5, 7 Zero: r when,,,,. r 5. Circle 5 Line 5. r sin Smmetric with respect to Circle with radius of. r cos 7. r cos. r sin Circle Radius:, center:, Cardioid Cardioid Houghton Mifflin Compan. All rights reserved. 9. r cos Limaçon. r sin Limaçon with inner loop. r 5 sin Limaçon

18 5 Chapter 9 Topics in Analtic Geometr. r cos. r 5 cos. r sin 5 Limaçon Rose curve Rose curve 5 5. r 7 sin. r cos 5 7. r cos Rose curve, four petals Rose curve, five petals 5 <. 9. r 5 sin. <.. r cos,. 5.. r cos,. r, sin cos r 9 sin r ±sin Graph both functions using. Houghton Mifflin Compan. All rights reserved.

19 Section 9.7 Graphs of Polar Equations 5 7. r sin cos,. 9. r csc < r e 5. r e Answers will var. Answers will var. 5. r cos,5. < 55. r cos, < 7 < r sin, < Use r sin and r sin. < Houghton Mifflin Compan. All rights reserved < r ± < 59. r sec is an asmptote.

20 5 Chapter 9 Topics in Analtic Geometr. r sin sin rr sin r sin r ± ± ± ± r csc sin ± The graph has an asmptote at. ± 5. r. is an asmptote.. True. It has five petals.. False. For eample, let r cos. 5. r cos5 n cos, < ; Answers will var. n 5 n n n n n n n Houghton Mifflin Compan. All rights reserved. n n n 5

21 Section 9.7 Graphs of Polar Equations 5. The graph of r f is rotated about the pole through an angle. Let r, be an point on the graph of r f. Then r, is rotated through the angle, and since r f f, it follows that r, is on the graph of r f. 7. Use the result of Eercise. (a) Rotation: Original graph: Rotated graph: (b) Rotation: Original graph: Rotated graph: (c) Rotation: Original graph: r f sin r f sin r f sin r f sin Rotated graph: r f sin f cos f cos r f sin f sin. (a) (b) (c) (d) r sin r sin r cos r sin r cos sin cos r sin r sin Houghton Mifflin Compan. All rights reserved. 9. (a) (b) (c) (d) r sin sin sin cos r sin sin sin sin cos r sin sin cos sin r sin sin sin sin cos 7. (a) (b) r sin r sin

22 5 Chapter 9 Topics in Analtic Geometr 7. r k cos k Circle k Conve limaçon k Cardioid k Limaçon with inner loop 7. r sin k (a) (b) (c) Yes. Answers will var. k.5: < k.5: < Section 9. Polar Equations of Conics The graph of a polar equation of the form r ep ± e cos Vocabular Check or r ep ± e sin is a conic, where e > is the eccentricit and is the distance between the focus (pole) and the directri. (a) If e <, the graph is an ellipse. (b) If e, the graph is a parabola. (c) If e >, the graph is a hperbola. Guidelines for finding polar equations of conics: (a) Horizontal directri above the pole: (b) Horizontal directri below the pole: r r (c) Vertical directri to the right of the pole: r (d) Vertical directri to the left of the pole: r ep e sin ep e sin ep e cos ep e cos. conic. eccentricit, e. (a) i (b) iii (c) ii p Houghton Mifflin Compan. All rights reserved.

23 Section 9. Polar Equations of Conics 55. r e e cos (a) Parabola (b) Ellipse (c) Hperbola a b c. (a) Parabola (b) Ellipse (c) Hperbola c a b. r e e sin (a) Parabola (b) Ellipse (c) Hperbola 9 b a c 9. (a) Parabola (b) Ellipse (c) Hperbola 9 c b a 9 5. r cos. r cos cos e parabola Vertical directri to left of pole Matches (b). e ellipse Vertical directri to left of pole Matches (c). 7. r cos cos. r sin 9. r sin e ellipse Vertical directri to right of pole e hperbola Horizontal directri below the pole. e hperbola Horizontal directri above the pole. Matches (f). Matches (e). Matches (d).. r sin. r cos. r sin Houghton Mifflin Compan. All rights reserved.. e Verte: Matches (a). r cos cos e ellipse, p, Vertices:, parabola r,,, 5, e parabola Verte: r,, e parabola Verte:,

24 5 Chapter 9 Topics in Analtic Geometr. r 7 7 sin 7 sin 5. r sin sin e 7 ellipse e ellipse Vertices: r, 7,, 7, Vertices: r, 7,,, 5. r cos cos 7. r sin sin e ellipse e ellipse Vertices:,, 5, Vertices:,,, 5. r 5 5 cos cos e hperbola Vertices: 5,, 5, 9. r cos cos e hperbola Hperbola Vertices: r,,,, 5.. r r 9 sin sin e hperbola Vertices: r, 5, 7 sin, 5, 7 sin 9. 9 r Parabola 5 sin 9. r Ellipse cos. 5 r Hperbola sin 5 Houghton Mifflin Compan. All rights reserved. Hperbola

25 Section 9. Polar Equations of Conics Ellipse Hperbola e,, p. Vertical directri to the left of the pole r cos cos e,, p Horizontal directri below the pole r sin sin 5. e,, p. e,, p Horizontal directri above the pole r sin sin Horizontal directri below pole ( r sin sin Houghton Mifflin Compan. All rights reserved. 7. e,, p. Vertical directri to the right of the pole r 9. Verte: cos cos Horizontal directri below the pole r, sin sin e,, p Vertical directri to the left of the pole r cos cos e, p. Parabola, e, verte:, Vertical directri to right of pole r ep e cos cos

26 5 Chapter 9 Topics in Analtic Geometr. Verte: Vertical directri to left of pole r. Center: Vertical directri to the right of the pole r r 5. Center:,, c, a, e p p cos cos p p cos 5 p 5 cos Vertical directri to left of pole r p r 5, e, p cos cos,, c, a, e c a p p cos cos cos p p. Verte: Horizontal directri above pole r. Center: Horizontal directri above the pole r p r. Center:,, c, a, e p p sin sin p sin sin Horizontal directri below the pole r p 9 5, 5,, c 5, a, e 5 5p 5p 5 sin 5 sin 5p 5 sin e, p sin sin 7. Center: 5,, c 5, a, e 5 Horizontal directri above the pole r Substitute the point rather than in order to get a directri between the vertices. p 5 r 5p 5p 5 sin 5 sin 5p 5 sin, 55 5 sin 5 sin, r sin 5 sin. Center: 5,, c 5, a, e c a 5 Horizontal directri above the pole r r 5p 5p 5 sin 5 sin 5p 5 sin p 5 5 sin Houghton Mifflin Compan. All rights reserved.

27 Section 9. Polar Equations of Conics When Therefore, Thus, r, r c a ea a a e. 5. Minimum distance occurs when. a e a e e ep a e ep. ep e cos ep e a. e cos e cos r e a e ea a e e cos e Maimum distance occurs when. r e a e ea a e e cos e 5. r cos cos Perihelion distance: r Aphelion distance: r a 5.9, e.5 r cos. 7.5 cos Perihelion distance: Aphelion distance: a e.55 7 miles a e. 7 miles 5. r cos cos Perihelion: r km Aphelion: r km 5. a.7 7, e.5 r cos cos Perihelion distance: Aphelion distance: a e.9 9 km a e.5 9 km Houghton Mifflin Compan. All rights reserved. 55. a.9 9, e., Neptune a 5.9 9, e., Pluto (a) Neptune: Pluto: r cos (b) Neptune: Perihelion: Aphelion: Pluto: Perihelion: r cos Aphelion: km km km km (d) Yes. Pluto is closer to the sun for just a ver short time. Pluto was considered the ninth planet because its mean distance from the sun is larger than that of Neptune. (e) Although the graphs intersect, the orbits do not, and the planets won t collide cos cos (c)

28 Chapter 9 Topics in Analtic Geometr ep 5. (a) Radius of earth miles. Choose r. e cos Vertices: a c 5,59.5 9,.5 e c a a Thus,,, and 9,, 9, ,59.5 ep e cos ep ep ep ep e cos e e e p a e e, r 5,9 ep 55.. Thus, r e cos (b) When and the distance from the surface of the earth to the satellite is 5,9,9 miles., r,7 5,59.5 (c) When and distance,7 miles cos 57. r sin sin False. The directri is below the pole. 5. r 9 cos False. The graph is not an ellipse. (It is two ellipses.) 59. r cos a r cos b r b cos r a r a cos a b r b a cos r a a b For an ellipse, b a c. Hence, r c cos r a a b r c a cos r b, e c a r e cos r b r e cos b r a. b r cos a r sin b b. e cos a b r cos a r sin b r cos a r cos b r b cos r a r a cos a b r b a cos r a a b a b c r c cos r a a b r c a cos r b, e c a r e cos r b r e cos b r b e cos b e cos Houghton Mifflin Compan. All rights reserved.

29 Section 9. Polar Equations of Conics. 9 a, b, c 5, e 5 r, 59 cos 9 5 cos. 9 r a, b, c 5, e 5 59 cos 5 cos 9. 5 a 5, b, c, e 5 r b e cos 95 cos 5 9 cos. a, b, c, e r b e cos 9 cos cos 9 5. Center:,,, c 5, a, e 5 b c a 5 9 r b e cos b 9 5 cos 5 cos. Center:,,, c, a 5, e 5 Houghton Mifflin Compan. All rights reserved. 7. b a c 5 9 r r b e cos Vertical directri to left of pole (a). cos e. ellipse 9 b 9 5 cos cos 9 (b) r Vertical directri to right of pole Graph is reflected in line r. cos. sin Horizontal directri below pole 9 rotation counterclockwise.

30 Chapter 9 Topics in Analtic Geometr. The lengths of the major and minor aes increase as p increases. Eample: r r.5.5 sin.5.5 sin 9. Answers will var. 7. r a sin b cos r a r sin b r cos a b Circle 7. tan 7. cos 7. sin 9 tan n cos 5 n, n sin sin ± n, n 7. 9 csc 75. cot 5 cos 7. sec csc csc cot sec cot sec sin sin ± n, n n cos 5 n, n For Eercises 77 : sin u cos u cos v sin v 5, 5,, cosu v cos u cos v sin u sin v sinu v sin u cos v sin v cos u sinu v sin u cos v sin v cos u cosu v cos u cos v sin u sin v Houghton Mifflin Compan. All rights reserved.. C 9. C 5. P 7. 9 P

31 Review Eercises for Chapter 9 Review Eercises for Chapter 9. Radius Radius Radius 5. Radius 5 5 Center 5,, Center, 5, 5.. Center: Radius:, Center: Radius:, Houghton Mifflin Compan. All rights reserved Center: Radius: Center: Radius: 7, 5 9 5, 9 9

32 Chapter 9 Topics In Analtic Geometr Center: Radius:, Center: Radius: 7, intercepts: 7. -intercepts: ±, -intercepts: 7 No -intercepts ± ±, impossible -intercepts: , impossible No -intercepts 5 7 ± ±, ±.. Verte: Focus:, p,, Directri:, p Verte: Focus:,, Directri: Verte: Focus: 9, p 9, 9, Directri: 9 Verte: Focus:, p,, Directri: 5 (, ) ), ) Houghton Mifflin Compan. All rights reserved.

33 Review Eercises for Chapter Verte:, Focus:, Parabola opens to left. p. Verte:, Focus:, Vertical ais, p 9. Verte:, Passes through, Vertical ais p p p. Verte:, 5 5 p p, on graph: 5 p p p 9 9., p. Focus:, d b p d 5 Focus:, Houghton Mifflin Compan. All rights reserved. d d Slope of tangent line: Equation: -intercept:, d (, b) F b 5 d (, ) b b Let b, be the -intercept of the tangent line. 7 b b m d b d 7 -intercept:,

34 Chapter 9 Topics In Analtic Geometr. p, on curve: p p p 9 9 if 9 width is meters.. (a) Parabola: 5. Verte: Passes through ±, Circle: Center: Vertices: Foci: p p p p Passes through ±, Radius: Center:, r, k k ± k k k, ± Eccentricit c a k a, b, c,, ± (b) Parabola: Circle:. d d d Center: Vertices: Foci: a, b, c 9, ±, ±, Eccentricit c a Houghton Mifflin Compan. All rights reserved.

35 Review Eercises for Chapter a, b, c 9 a, b, c Center: Vertices: Foci:,, ± Eccentricit c a,,, Center:, Vertices:,, 5, Foci: ±, Eccentricit c a (a) 9 (c) 9 5 (b) Center:, 9 5 a, b, c 9 7 Vertices:,,, Foci:, ± 7 e c a 7. (a) (c) Houghton Mifflin Compan. All rights reserved. (b). (a) a 5, b, c Center: Vertices:,, 7, Foci: ±, e 5 5 5, 9 9 CONTINUED 7 7

36 Chapter 9 Topics In Analtic Geometr. CONTINUED (b) Center:, 7 (c) a, b c a b 5 Vertices: ±, 7 Foci: ±, 7 c Eccentricit: c a. (a) 5 5 (b) a 5, b, c 5 9 (c) Center: e c a 5, Vertices: 5 ± 5, Foci: 5 ± 9, Vertices: Foci: ±, ±5, a 5, c b 5 9. Vertices: Passes through, Vertical major ais Center:, ±,, a b b b 9 9 b 9 9 Houghton Mifflin Compan. All rights reserved.

37 Review Eercises for Chapter Vertices: Foci: Horizontal major ais Center: a 5, c, b 5 h a,, 7,,,,, 5 k b. Vertices: Foci: Vertical major ais Center: a, c, b h b,,,,,,, k a 7. a 5, b, c a b 5. The foci should be placed feet on either side of the center and have the same height as the pillars., a, b 9 9 Longest distance: Shortest distance: c a b Foci: ±, Distance between foci: a feet b feet. feet 9. a c a c.55 9 Adding, Then c e c.5. a a.5 9 a.7 9. a 7 e c.5 c ae 7. a b a c Houghton Mifflin Compan. All rights reserved.. (a) (b) 5 a, b 5, c 5 Center: Vertices: Foci: 5,, ±, ± Eccentricit c a (c)

38 7 Chapter 9 Topics In Analtic Geometr. (a) (b) a, b, c Center: Vertices: Foci:, ±, ±, (c) Eccentricit c a. (a) (c) 9 (b) Center:,, a, b, c 5 Vertices: 5,,, 9 Foci:,,, Eccentricit: 5. (a) (c) (b) Center: Vertices: Foci:,, a, b 5, c 9, ± 9 Eccentricit: 9 5,,, 5 5 Houghton Mifflin Compan. All rights reserved.

39 Review Eercises for Chapter (a) 59 (b) Center: Foci:, Eccentricit: e c a a, b, c 55 Vertices:, ± ± 55, 55 5 (c). (a) (b) Center:, (c) a, b, c Vertices: ±, :,, 5, Foci: ±, Eccentricit: e Houghton Mifflin Compan. All rights reserved. 7. a b a c a b b b 5. Vertices: Foci: Vertical transverse ais Center: a, c, b 9, ±,, ±

40 7 Chapter 9 Topics In Analtic Geometr 9. Foci: Center: Asmptotes: ± b b a a c a b a a 5a a b 5, 5,,, c, Vertical transverse ais Center: a a b b b b k a 5, c a b c b 5 a 5 h b 5 5. d d,.5 a 9 a.5 c b c a a b (, ) B (, ) A (, ) b a miles north 5. Let the friends be at B and C, ou at the origin A. The sound at C is heard seconds after B: a CD BD 5 5 Thus, a 5 and b c a 79., c 57. Thus, using miles, the hperbola is Now place the center at, and determine the second hperbola. a DB AD 5 a 5 c and b C A D B C A D B Houghton Mifflin Compan. All rights reserved.

41 Review Eercises for Chapter Ellipse A C Circle Hperbola A, C, AC Parabola 57. A, B, C cot A C B, 5 5 Houghton Mifflin Compan. All rights reserved. 5. Hperbola A C cot, Hperbola 5

42 7 Chapter 9 Topics In Analtic Geometr A 5, B, C 5 cot, Ellipse.. (a) cot, 7 9 Parabola (b) (c) 7 9 B AC. (a) B AC Ellipse Parabola 5 5 ± 5 5 (b) (c) 7 5 ± 7 5 Houghton Mifflin Compan. All rights reserved.

43 Review Eercises for Chapter (a) B AC (c) (b) Parabola ± 5. (a) (b) (c) B AC ± ± 9 5 ± 9 > Hperbola 5. Adding the equations,. Then: 5 Solution:,. 9 Adding: t , 5 If : Houghton Mifflin Compan. All rights reserved ± If 5, 9 5, impossible. Answer:,,, t, t t

44 7 Chapter 9 Topics In Analtic Geometr 9. 5t, t 5 7. t , line t, t t t, t 7. ln t, t t e t, t e e e, t, t 7. t t 7. t t t 5 t, t t t 77. t t t t Houghton Mifflin Compan. All rights reserved.

45 Review Eercises for Chapter t 79. t. t, t t t t t, Line, 5. t t t. t, t 7. t t Vertical line: 9. t 7 5. cos, sin cos, sin Houghton Mifflin Compan. All rights reserved.. cos, 5 sin cos, sin t, t t, t Other answers possible

46 7 Chapter 9 Topics In Analtic Geometr. t, t 9. t, t Man answers possible t, t t, t t t Other answers possible 9. t, t 5t 9. t, t 5t Man answers possible t t 5t t 5 t 5 or t, 5 9. t t 9. t t 5t or, t t t t t t t 9. t t 5 5 t t t or 5t, t t 95. 9, is on the curve: 9.v t v 9.t t t t t.579. Hence, v t ftsec From Eercise 95, v From Eercise 9: t.7t t t 7.9t t (, ( 5 The maimum height is approimatel.9 feet for t.97.,. ( 5, ( 9. From Eercise 95, t. seconds. 5 Houghton Mifflin Compan. All rights reserved., 7, 5,,, 5, 5, 5,, 5,

47 ) Review Eercises for Chapter r,,. r,, 5 5 (, ), ),,, 7,, 5,,,,, 7. 5,. (, ) ( 5, ( 5,, 5,, 5 5, r,, 5,,, 7, 5. r, 5, 7., 5, 5 Houghton Mifflin Compan. All rights reserved. 7. ( 5, 7, 5 ( r cos r sin,, ) 5, ) 5, r,, ( (, cos, sin,

48 Chapter 9 Topics In Analtic Geometr., 9. r cos r sin,, ), ) r,,, cos, sin (, (,. r,, the origin,.,,,, 9 r, 9,, 9, (, ( (, 9)..,,. r 5, tan 5,.7, 5,.7 or 5,., 5, 5.5 (radians),, Third quadrant, 7 r r,, 7,, (, ) r (, (, 5, 5 r, 5, 7, 5 5 (5, 5) 5, Houghton Mifflin Compan. All rights reserved.

49 Review Eercises for Chapter r 9 r r r cos r r sin r r 5 r cos r sin 9. r cos r sin 5 5. r 5 csc sec r cos r sin r sec csc. r cos r sin r cos r cos. r cos r cos r cos r sin r cos sin r sin sin r sin r sin. r r 5. r cos r r cos Circle Circle. r sin r r sin 7. r cos r sin r r r sin Houghton Mifflin Compan. All rights reserved.. r sin r r sin or 9. 5 tan, line. tan. r 5, circle. r, circle. -ais,

50 Chapter 9 Topics In Analtic Geometr., line 5. r 5 cos,. circle r sin, circle 5 7. r 5 cos Dimpled limaçon Smmetric with respect to polar ais is maimum at r (No zeros) : r r, 9,. r sin Limaçon with inner loop Smmetric with respect to r is a maimum at : 5, r when sin sin.9,. 9.. r 5 sin Limaçon with loop Smmetr: line r Maimum r-value: when.5,.9 Zeros: r when sin 5 r cos Limaçon with inner loop Smmetr: polar ais Maimum: r when Zero: r when cos., Houghton Mifflin Compan. All rights reserved.

51 Review Eercises for Chapter 9. r cos, Four-leaved rose Smmetric with respect to polar ais, and pole, The value of is a maimum () at,, r. r for,, 5, 7,. r cos 5 Five-leaved rose Smmetric with respect to polar ais r is maimum value of at r for n n, n,,,..., n,,, r 5 sin Lemniscate Smmetr with respect to pole Maimum r-value: 5 when Zeros: r when,,,, 5 Houghton Mifflin Compan. All rights reserved.. 5. r cos Lemniscate Smmetr: Pole, polar ais, and line Maimum: r when Zeros: r when,, 5, 7 r e Parabola sin,,. r Hperbola smmetric with and having vertices at, and,, e sin

52 Chapter 9 Topics In Analtic Geometr 7. r 5 cos. r cos cos 5 5 cos Hperbola e e 5 Ellipse 9. r 5 sin 5. r cos cos 5 sin Parabola e e 5. Ellipse e r cos Vertical directri: ep 5. Parabola: r, e e sin Verte:, Focus:, p r sin 5. Ellipse: Vertices: One focus: p e c 5 cos p 5 a, r r ep e cos 5,,, a, c 5 cos ep 5. Hperbola: r e cos 5 cos Vertices:,, 7, a One focus:, c p e c cos p 7 a, 5 cos 7 7 r 7 cos cos cos Houghton Mifflin Compan. All rights reserved.

53 Review Eercises for Chapter e.9 ep Use r. e cos a r.5.9 cos Perihelion: Aphelion:.9p.9 cos.9p.7p.5 p.5, ep.5.9 cos astronomical units astronomical units ep 5. Use r (horizontal directri below pole). e sin e When r When (parabola) p sin r,, sin, r,,. p,, p,, distance is approimatel,,7 miles., Houghton Mifflin Compan. All rights reserved. 57. False. The -term is not second degree.. (a) Major ais horizontal (b) Circle (c) Ellipse is flatter. (d) Horizontal translation 5. False. There are man sets possible. For eample, t, t t, t. 59. (a) Vertical translation (b) Horizontal translation (c) Reflection in the -ais (d) Parabola opens more slowl.. The number b must be less than 5. The ellipse becomes more circular and approaches a circle of radius 5.. The orientation of the graph would be reversed.. (a) The speed would double. (b) The elliptical orbit would be flatter. The length of the major ais is greater.

54 Chapter 9 Topics In Analtic Geometr Chapter 9 Practice Test. Find the verte, focus and directri of the parabola.. Find an equation of the parabola with its verte at, 5 and focus at,.. Find the center, foci, vertices, and eccentricit of the ellipse.. Find an equation of the ellipse with vertices, ± and eccentricit e. 5. Find the center, vertices, foci, and asmptotes of the hperbola.. Find an equation of the hperbola with vertices at ±, and foci at ±5,. 7. Rotate the aes to eliminate the -term. Sketch the graph of the resulting equation, showing both sets of aes Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hperbola. (a) (b) 7 For Eercises 9 and, eliminate the parameter and write the corresponding rectangular equation. 9. sin, 5 cos. e t, e t. Convert the polar point, to rectangular coordinates.. Convert the rectangular point, to polar coordinates.. Convert the rectangular equation to polar form.. Convert the polar equation r 5 cos to rectangular form. 5. Sketch the graph of r cos.. Sketch the graph of r 5 sin. 7. Sketch the graph of r. cos. Find a polar equation of the parabola with its verte at, and focus at,. Houghton Mifflin Compan. All rights reserved.