C H A P T E R 9 Topics in Analytic Geometry

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1 C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas Section. Ellipses Section. Hperbolas Section. Rotation and Sstems of Quadratic Equations Section. Parametric Equations Section. Polar Coordinates Section.7 Graphs of Polar Equations Section. Polar Equations of Conics Review Eercises Practice Test Houghton Mifflin Compan. All rights reserved.

2 C H A P T E R Topics in Analtic Geometr Section. Circles and Parabolas A parabola is the set of all points, that are equidistant from a fied line (directri) and a fied point (focus) not on the line. The standard equation of a parabola with verte h, k and (a) Vertical ais h and directri k p is ( h) p( k), p 0. Horizontal ais k and directri h p is ( k) p( h), p 0. The tangent line to a parabola at a point P makes equal angles with (a) the line through P and the focus. the ais of the parabola. Vocabular Check. conic section. locus. circle, center. parabola, directri, focus. verte. ais 7. tangent Radius 7 0. Radius h k r h k r 7 Diameter 7 radius 7. Diameter radius h k r h k r , 0 0, 0, 7 Radius: 7 Radius: Radius: Houghton Mifflin Compan. All rights reserved. 77

3 Section. Circles and Parabolas ,, 0 0, Radius: Radius: Radius:... 0, 0 Radius: 0, 0 Radius: 0, 0 Radius:. 7., 0, 0 Radius: Radius:. 0., Radius: Radius:, Houghton Mifflin Compan. All rights reserved.. Radius: 0 Radius:, 0, Radius: 0,

5 Section. Circles and Parabolas intercepts: 0. -intercepts: Let 0. 0, -intercepts: 0, 0,, 0 0 ±, 7 0 0,, 0, -intercepts: Let 0. 7 ±, ± ± 7 ± 7, 0 Houghton Mifflin Compan. All rights reserved.. -intercepts: Let 0.. -intercepts: 0 0 No solution No -intercepts -intercepts: Let ±7 ± 7 ± 7, 0 No solution No -intercepts No solution No -intercepts -intercepts: No solution No -intercepts -intercepts: 0 ± 7, 0. -intercepts: 0 7. (a) Radius: ; 0, ±7 ± 7 The distance from 0, to 0, 0 is 0 7 miles. Yes, ou would feel the earthquake = (0, ) You were 7 miles from the outer boundar.

6 77 Chapter Topics in Analtic Geometr. (a) Area r 00 r 00 r 00 r.7 feet R 00 R feet longer radius 7... Verte: 0, 0 Verte: 0, 0 Verte: 0, 0 Opens to the left since p is negative. Matches graph (e). p > 0 Opens upward Matches graph. Opens downward since p is negative. Matches graph (d). 0.. ( ) ( ). Verte: 0, 0 Verte:, Verte:, p < 0 Opens to the left Matches graph (f). Opens to the right since p is positive. Matches graph (a). p < 0 Opens downward Matches graph.. Verte: Graph opens upward. p Point on graph: p p p 0, 0 h 0, k 0, Thus,. Focus: p 0. Verte: Directri:., 0 p 0, 0 h 0, k 0 p h p k 0 0 or. Point: a a a 7. Verte:, Focus: 0, 0 h 0, k 0, 0 p k p h 0. Directri: p p. Verte: 0, 0 h 0, k 0 Focus: 0, p h p k. Focus: 0, p p. Verte: 0, 0 h 0, k 0 Directri: p p Houghton Mifflin Compan. All rights reserved.

7 Section. Circles and Parabolas 777. Directri: p p. Verte: 0, 0 h 0, k 0 Horizontal ais and passes through the point, k p h 0 p 0 p p p p. Vertical ais Passes through, p p p p.. 7. ; p Verte: Focus: Directri: 0, 0 0,, p Verte: Focus: Directri: 0, 0 0, ; p Verte: Focus: Directri: 0, 0, 0 Houghton Mifflin Compan. All rights reserved... ; p Verte: Focus: Directri: 0, 0, ; p Verte: 0, 0 Focus: 0, Directri: 0, p Verte: Focus: Directri: 0, 0, 0

8 77 Chapter Topics in Analtic Geometr. 0. h, k, p Verte:, Focus:, Directri: 0 0 Verte: Focus: Directri:,,, ; p ; p Verte:, Verte:, Focus:, Directri: 0 0 Focus: 0, Directri:. h, k, p. Verte:, Focus:,, p,,, Verte: Focus: Directri: Directri: 0 7. h, k, p Verte: Focus:,, Directri: 0. 0 Verte:, Focus:, Directri: 7 0 Houghton Mifflin Compan. All rights reserved.

9 Section. Circles and Parabolas Verte:, Focus:,, Directri: 0 Verte: Focus:,, Directri: h, k, p Verte: Focus: Directri:, 0, To use a graphing calculator, enter: 0 Verte:, 0 Focus: 0, 0 Directri: Houghton Mifflin Compan. All rights reserved. 7. Verte: Passes through:,, opens downward 7. Verte: Focus:, 0,, 0, h,, p h p k k 7. Verte: Passes through:, h, k k p h p p. p., 77. Verte:, Focus:, 7. Verte: Focus: p, 0, opens to the right, 0 Horizontal ais: p

10 70 Chapter Topics in Analtic Geometr 7. Verte: Focus:, h, k, 0 p h p k 7. Verte: 0, Directri: Vertical ais p 0 0. Verte: Directri:, h, k p k p h. Focus:, Directri: Horizontal ais Verte: 0, p 0 0. Focus: 0, 0 Directri: p Verte: 0,. 0 and 0. and The point of tangenc is,. 0 and 0 The point of tangenc is,..,, p, focus:, 0, Following Eample, we find the -intercept 0, b. d b d 0 7 d d m, Let 0 7 b Tangent line b 0 -intercept, 0. Houghton Mifflin Compan. All rights reserved.

11 Section. Circles and Parabolas 7. p Focus: d b d 0 b m 7. 0, b 0 Focus: Following Eample, we find the -intercept 0, b. d b d 0 7 d d m 0 p 0, b 7 b Let 0 intercept -, 0. Tangent line: 0 -intercept:, 0.,,. p Focus: 0, R 7 R is a maimum of $,7.0 when televisions.,000 Houghton Mifflin Compan. All rights reserved. d b d 0 d d b b m 0 Intercept:,

12 7 Chapter Topics in Analtic Geometr 0. (a) p p 0 p 07 p 07,,. feet. (a) p, p or When, in. (0, (,. Depth: inches. p,, on parabola. (a) p p The wire should be inserted inches from the bottom. p 0 (, ) ( 0, ) (0, ) 0 p p,00, (a) p passes through point,.. (a) p p or feet V 7,00 mihr,70 mihr 7. p 00, h, k 0, , Verte: p Point: 0, 0 000, p000 (a) 0 0 p The highest point is at., 7.. The distance is the -intercept of. feet. Houghton Mifflin Compan. All rights reserved.

13 Section. Circles and Parabolas 7. (a) v s. The slope of the line joining, and the center is. The slope of the tangent line at, is. Thus, s v 7 7, tangent line ft 00. The slope of the line joining, and the 0. The slope of the line joining, and center is. The slope of the tangent line the center is. The slope of at, is Thus, the tangent line is. Thus,. 0, tangent line., tangent line. 0. The slope of the line joining, and the center is. The slope of the tangent line is. Thus, 0 0, tangent line. 0. False. The center is 0,. 0. True 0. False. A circle is a conic section. Houghton Mifflin Compan. All rights reserved. 0. False. A parabola cannot intersect its directri or focus. 0. Answers will var. See the reflective propert of parabolas, page. 0. The graph of 0 is a single point, 0, True 0. False. The directri is below the -ais. The plane intersects the double-napped cone at the vertices of the cones.

14 7 Chapter Topics in Analtic Geometr.. For the upper half of the parabola,. For the lower half of the parabola,.. f. f Relative maimum: 0.7,.7 Relative minimum:. at 0.7 Relative minimum: 0.7, 0.. f. Relative minimum: 0.7, 0. f Relative minimum:. at 0. Relative maimum:. at 0. Section. Ellipses An ellipse is the set of all points, the sum of whose distances from two distinct fied points (foci) is constant. The standard equation of an ellipse with center h, k and major and minor aes of lengths a and b is (a) h a h b k b k a if the major ais is horizontal. if the major ais is vertical. c a b where c is the distance from the center to a focus. The eccentricit of an ellipse is e c a. Vocabular Check. ellipse. major ais, center. minor ais. eccentricit... 0, 0 0, 0 0, 0 a, b Vertical major ais Matches graph. a, b Horizontal major ais Matches graph. a, b Vertical major ais Matches graph (d). Houghton Mifflin Compan. All rights reserved.

15 Section. Ellipses 7.. 0, 0 a, b Horizontal major ais Matches graph (f).., a, b Horizontal major ais Matches graph (a)., Horizontal major ais Matches graph (e) , 0 0, 0 a, b, c Vertices: ±, 0 Foci: ±, a, b, c Vertices: 0, ± Foci: 0, ± e c a e c a. a, b, c Vertices: Foci:,, ± ;,,,, ± ;,,, 0. Foci: Vertices:, a, b, c, ± ;, 0,,, ± ;,,, e c a e c a Houghton Mifflin Compan. All rights reserved.. Foci: Vertices:, a, b, c 0,,,, 7,,,, 7 7 e

16 7 Chapter Topics in Analtic Geometr. a, b, c a b, Foci:,,, Vertices:,,, Eccentricit:. (a) 0 a, b, c 0, 0 Vertices: ±, 0 Foci: ±, e c a. (a) a, b, c 0, 0 Vertices: 0, ± Foci: 0, ± e c a. (a) 0 a, b, c Foci:,, ± Houghton Mifflin Compan. All rights reserved. Vertices:,,, 0 e

18 7 Chapter Topics in Analtic Geometr. (a) 0 0 ( ) ( ) a, b, c, Foci: Vertices: 7,,,,,, e 0. (a) Degenerate ellipse with center, as the onl point. (a) a, b, c Vertices: Foci: Eccentricit:, ±, ±, c 0 a Houghton Mifflin Compan. All rights reserved.

19 Section. Ellipses 7. (a) 0 a, Vertices: Foci: Eccentricit: b c,,, ± c a, ±,,,. a, b Vertical major ais 0, 0. Vertices: Endpoints of minor ais: 0, ± b a b ±, 0 a Houghton Mifflin Compan. All rights reserved.. a, c b Horizontal major ais 7. 0, 0 c 0, 0 a b 7 Horizontal major ais 7. Vertices: Foci: b a c k a. c 0, ± a 0, ± c 0, 0 h, k 0, 0 h b a b Horizontal major ais

20 70 Chapter Topics in Analtic Geometr. Vertices: Vertical major ais h b Point: b , 0, 0, ± a b 00 b 00 k a b b 0. Vertical major ais Passes through: a, b b a 0, and, 0. Vertical major ais h b, a, b. k a a, b c Horizontal major ais., 0, c, a b a c 0 Vertical major ais 0. Vertices: Endpoints of minor ais: h a. 0,,, a, h, k k b Horizontal major ais, 0, 0,, b c, a b a c Houghton Mifflin Compan. All rights reserved.

21 Section. Ellipses 7. Verte: Minor ais length: h b. a c Foci: h a, h, k, a k a, h, k b,,, c, a b a c k b 7. Vertices: Minor ais of length b Vertical major ais h b,. Vertices:,,, a k a Horizontal major ais h a 0,,,, a a c c c b b k b Houghton Mifflin Compan. All rights reserved. 0. Vertices:, 0,, a. Endpoints of minor ais: 0,, 0, b h b 0 a, b, c e c a, h, k 0 0 k a. a, b, c e c a. a, b, c e c a

22 7 Chapter Topics in Analtic Geometr. 0 7 a, b, c e c a. Vertices: Eccentricit: b a c 0, 0 Horizontal major ais ±, 0 a c a c a. Vertices: Eccentricit: b a c b a 0, ± a, h 0, k 0 e c a c c 7. (a). (a) ( 0, 0) (0, 0) (0, 0) (0, ) (, 0) (, 0) Vertices: Height at center: 0 b 0 Horizontal major ais a b ±0, 0 a 0, a, b, 0. Let be the equation of the ellipse. Then b and a b a c a b. Thus, the tacks are placed at ±, 0. The string has a length of a feet. For, The height five feet from the edge of the tunnel is approimatel 7. feet. When 0, 0. >. Hence, the truck will be able to drive through without crossing the center line. Houghton Mifflin Compan. All rights reserved.

23 Section. Ellipses Area of ellipse area of circle 0 ab r a a or 7 a 0 Length of major ais: a 0 0 units a 7, b, c 7.7 Distance between foci:.7. feet. a. a 7. a. e c a c a b b a c.0 Ellipse: 0, 0, e c c 7.0 a c.0 a c 0. a. a. c.7 b a c b a c For, we have c a b. a b a c 7 0 When c, a, a c c a. b a b b a b a c 7.. a c a b a Houghton Mifflin Compan. All rights reserved.. e c 0.0 a a, b, c Points on the ellipse: ±, 0, 0, ± Length of latus recta: b a ( (,, a + c 7. ) ) b ( (, ), ) Points on the ellipse: ±, 0, 0, ± Length of latus recta: b a b a. a, b, c 7 ( ), 7, 7,, 7 7 ( ) ( ) ( ) Additional points:, ±,, ± Additional points: ±, 7, ±, 7

24 7 Chapter Topics in Analtic Geometr.. (, (, ) ), ) ( Points on the ellipse: a, b, ±, 0, 0, ± (, ) ( c (, (, ) ) Length of latus recta: Points on the ellipse: b ±, 0, 0, ± a b Length of latus recta: a Additional points: ±,, ±, (, ), Additional points: ±,, ±, ) 0. Answers will var.. True. If e then the ellipse is elongated, not circular.. True. The ellipse is inside the circle.. (a) The length of the string is a. The path is an ellipse because the sum of the distances from the two thumbtacks is alwas the length of the string, that is, it is constant.. (a). Foci: a b 0 b 0 a A ab a0 a a 0a 0 a or a b the Quadratic Formula b ORb Since a > b, we choose a and b. Horizontal major ais a0 a (d),.,, 0, c a c a c a a b a b b 0 0 a A The area is maimum when a b 0 and it is a circle. a b The sum of the distances from an point on the ellipse to the two foci is constant. Using the verte a, 0, ou have a c a c a. From the figure, b c a a b c. a c b b b c b + c c a Houghton Mifflin Compan. All rights reserved.

25 Section. Hperbolas 7 7. Arithmetic: d. Geometric: r. Geometric: r 70. Arithmetic: d 7. n 0 7. n n0 n0 n n n0 n 0. Section. Hperbolas A hperbola is the set of all points, the difference of whose distances from two distinct fied points (foci) is constant. The standard equation of a hperbola with center h, k and transverse and conjugate aes of lengths a and b is: (a) h a k a k b h b if the transverse ais is horizontal. if the transverse ais is vertical. c a b where c is the distance from the center to a focus. The asmptotes of a hperbola are: (a) k ± b h a k ± a h b if the transverse ais is horizontal. the transverse ais is vertical. The eccentricit of a hperbola is e c a. To classif a nondegenerate conic from its general equation A C D E F 0: (a) If A C (A 0, C 0), then it is a circle. If AC 0 (A 0 or C 0, but not both), then it is a parabola. If AC > 0, then it is an ellipse. (d) If AC < 0, then it is a hperbola. Houghton Mifflin Compan. All rights reserved. Vocabular Check. hperbola. branches. transverse ais, center. asmptotes. A C D E F 0. 0, 0. 0, 0 a, b, c a, b Vertical transverse ais Matches graph. Vertical transverse ais Matches graph.., 0., a, b a, b Horizontal transverse ais Matches graph (a). Horizontal transverse ais Matches graph (d).

26 7 Chapter Topics in Analtic Geometr.. a, b, c 0, 0 Vertices: ±, 0 Foci: ±, 0 Asmptotes: ± 0, 0 a, b, c Vertices: ±, 0 Foci: ±, Asmptotes: ± b a ± 7.. a, b, c a, b, Vertices: Foci: 0, 0 0, ± 0, ± Asmptotes: ± c 0 Vertices: Foci: 0, 0 0, ± 0, ±0 Asmptotes: ±. 0. a, b, c a b 0 Vertices: Foci: 0, 0 0, ±0 Asmptotes: 0, ± a, b, c 0 0, 0 Vertices: ±, 0 Foci: ±0, 0. ± a b ± Vertices:,,, Foci: a, b, c, ±, Asmptotes: ±. Asmptotes: ± Vertices: Foci: Asmptotes:, a, b, c,,,,, 0, ± Houghton Mifflin Compan. All rights reserved.

27 Section. Hperbolas 77. a, b, c Vertices: Foci:, Asmptotes:, ± :,,,, ±. k ± a h b ± a, b, c Vertices: Foci:, Asmptotes:,,,, ± ± ±. (a) 0, 0 a, b, c Vertices: ±, 0 Foci: ±, 0. (a) 00 0, 0 a, b, c Vertices: ±, 0 Foci: ±, 0 Houghton Mifflin Compan. All rights reserved. Asmptotes: ± b a ± Asmptotes: ± b a ±

28 7 Chapter Topics in Analtic Geometr 7. (a) To use a graphing calculator, solve first for. a, b, c 0, 0 Vertices: ±, 0 Foci: ±, 0 Asmptotes: ± ± Hperbola Asmptotes. (a) a, b, c Vertices: 0, 0 0, ± Foci: 0, ± Asmptotes: ± ±. (a) 0 0. (a) a, b, c 0 Vertices: Foci:,,,, ± 0, Asmptotes: ± a, b, 0, Vertices: ±, Foci: ±0, c 0 Houghton Mifflin Compan. All rights reserved. Asmptotes: ±

29 Section. Hperbolas 7. (a) ± Degenerate hperbola is two lines intersecting at,.. (a) 0 0 ± Degenerate hperbola is two intersecting lines at,.. (a) Vertices: Foci: 0,, ±, ± a, b, c Asmptotes: ± To use a graphing calculator, solve for first. Houghton Mifflin Compan. All rights reserved. ± Hperbola Asmptotes 0

30 00 Chapter Topics in Analtic Geometr. (a) a 0, b, c 0 7, Vertices: ±, Foci: 0 ±, Asmptotes: ±. Vertices: Foci: k a 0, ± a 0, ± c b c a 0, 0 h, k h b. Vertices: Foci: a b 7 ±, 0 a ±, 0 c b c a 7 7. Vertices: Asmptotes: ±, 0 a ± b a b 0, 0. Vertices: 0, ± a. Foci: 0, ± c Asmptotes: ± a, b b Asmptotes: ± a a b b k a 0, 0 h, k h b c a b b b 7 b a 0 7 k a 0, 0 h, k h b Houghton Mifflin Compan. All rights reserved.

31 Section. Hperbolas 0 0. Foci: Asmptotes: ±0, 0 c 0 c a b 00 m m a, b a b ± b a 00 m m m m. Vertices: Foci: b c a h a, 0,, 0 a 0, 0,, 0 c, 0 h, k k b. Vertices: Foci: b c a k a,,, c,,, a, 0 h b. Vertices: Foci: b c a k a,,, a, 0,, 0 c, h, k h b Houghton Mifflin Compan. All rights reserved.. Vertices: Foci: b c a h a,,, c,,, a 0, k b. Vertices: Solution point: k a,,, a 0,, 0 h, k h b b b

32 0 Chapter Topics in Analtic Geometr. Solution point: 0,, a b b 7, b b 7 7. Vertices: Passes through, b 0,, 0, 0 0,, a b b b b. Solution point:, 0, a b b 0, b b. Vertices:,,, a Asmptotes:,, b b a 0. Asmptotes:,, a, a b b b. Vertices: 0,,, a. Vertices: (, 0,, a Asmptotes:, Asmptotes:, b a b h a, h, k k b a b b k a, h, k h b Houghton Mifflin Compan. All rights reserved.

33 Section. Hperbolas 0. F : Friend s location 0,0, 0 F : Your location 0,0, 0 P, : a b Location of lightning strike 00,00 c 0,0, a,00 b c a,0,00,00,000,0,00 00 a,00,000 0,000 P Friend F F You 0,000 0,000 ( 0,0, 0) (0,0, 0) 0,000. The eplosion occurred on the vertical line through (00, 00) 00, 00 and 00, d d Hence, a 00 a 00 c 00 b c a. The eplosion occurred on the hperbola Letting 00, b a , 70 a. b ( 00, 0) d a (00, 0) d Houghton Mifflin Compan. All rights reserved.. (a) a b a ;, is on the curve, so b b b, 7 b. Because each unit is foot, inches is of a unit. The base is units from the origin, so. When, 7.. So the width is.77 units, or. inches, or. feet.

34 0 Chapter Topics in Analtic Geometr. Foci: ±0, 0 c 0 0, (, 7) d d ( 0, 0) (0, 0) 0 (a) d d, a a b c a 0,, 0. miles 0 7 miles 7,., Ba to Station : 0 miles Ba to Station : 70 miles 70 0,000 (d) In this case, d d, a 0 and b c a 00. The hperbola is 0. 0 For 0, 0,00 and.. Position:., second 7.. Focus: 0, 0, 0 b c a a 7 a a 7 a a 7 a 7 7 a 7 a 77 a 7a a 7 a a 7a,77 0 a ±. or a ±. Since a < c and c, we choose a.. The verte is approimate at., 0. [Note: B the Quadratic Formula, the eact value of a is a. ]. a, b, c The camera is units from the mirror. 0 A, C AC > 0, Ellipse Houghton Mifflin Compan. All rights reserved.

35 Section. Hperbolas A C, Circle 0 0 A, C AC < 0, Hperbola A, C 0 AC 0, Parabola 0 C, A 0 AC 0, Parabola A, C AC 00 > 0, Ellipse 0 A C, Circle A, C AC < 0, Hperbola 7 0 A, C 0, D, AC 0 Parabola E, F True. e c a a b A, C AC > 0 Ellipse a 0. False. b 0 because it is in the denominator. Houghton Mifflin Compan. All rights reserved.. False. For eample, 0 0 is the graph of two intersecting lines.. True. The asmptotes are ± b a. If the intersect at right angles, then b a ba a b. Let, be such that the difference of the distances from c, 0 and c, 0 is a (again onl deriving one of the forms). a c c a a c c c a c c a a c c a a c c a c c c a c a a c a c a a a b. Let b c a. Then a b b a a b.

36 0 Chapter Topics in Analtic Geometr. Answers will var. See Eample.. d d constant b definition of hperbola At the point a, 0, d d a c c a a.. Horizontal transverse ais Foci at, and 0, c. c a c a a b c a 7, 7 7. At the point a, 0, the difference of the distances to the foci ±c, 0 is c a c a a. Let, be a point on the hperbola. a c c a c c a a c c c a c c a a c c a a c c c a c a a c a c a a Thus, c a b, as desired. a c a. If A C 0, then b completing the square ou obtain a circle. If A 0 and C 0, then C D E F 0 is a parabola (complete the square). Same for A 0 and C 0. If AC > 0, then both A and C are positive (or both negative). B completing the square ou obtain an ellipse. If AC < 0, then A and C have opposite signs. You obtain a hperbola Houghton Mifflin Compan. All rights reserved.

37 Section. Rotation and Sstems of Quadratic Equations i i Section. Rotation and Sstems of Quadratic Equations The general second-degree equation A B C D E F 0 can be rewritten A C D E F 0 b rotating the coordinate aes through the angle where cot A CB. cos sin sin cos The graph of the nondegenerate equation A B C D E F 0 is: (a) An ellipse or circle if B AC < 0. A parabola if B AC 0. A hperbola if B AC > 0. Vocabular Check. rotation, aes. invariant under rotation. discriminant Houghton Mifflin Compan. All rights reserved. 0;. Point: cos sin 0 cos 0 sin 0 0 0, Thus,,, 0. ;. Point:, cos sin cos sin sin cos sin 0 cos 0 Adding,. sin cos sin cos Subtracting, 0 0. Thus,,, 0.

38 0 Chapter Topics in Analtic Geometr. 0 A 0, B, C 0 cot A C B cos sin sin cos 0 0 0, Hperbola. 0, A 0, B, C 0 cot A C B cos sin sin cos , Hperbola A, B, C sin cot A C 0 B cos cos sin Houghton Mifflin Compan. All rights reserved.

40 0 Chapter Topics in Analtic Geometr 7. 0 A 0, B, C 0 cot A C B 0 cos sin sin cos , Hperbola. 0 0 A, B, C cot C B 7.7 cos sin cos 0 cos cos 0 cos sin sin cos CONTINUED Houghton Mifflin Compan. All rights reserved.

41 Section. Rotation and Sstems of Quadratic Equations. CONTINUED , Hperbola. 0 A, B, C cot A C B cos sin 0 sin cos 0, Ellipse Houghton Mifflin Compan. All rights reserved A, B, C 7 cot A C B cos sin CONTINUED sin cos

43 Section. Rotation and Sstems of Quadratic Equations A, B, C cot C B 7. cos 7 sin cos 7 cos cos 7 cos sin sin cos Parabola Houghton Mifflin Compan. All rights reserved A, B, C cot A C B cos 7 sin cos 7 cos cos 7 cos sin 7. sin cos CONTINUED

45 Section. Rotation and Sstems of Quadratic Equations. 0 cot A C B Solve for in terms of : 0 0 Graph 0 and ± Houghton Mifflin Compan. All rights reserved.. 7. A, B, C cot A C B tan tan To graph conic with a graphing calculator, we need to solve for in terms of. Graph ±. ± and cot A C B Solve for in terms of b completing the square. 7 Graph ± 7 ± 7 and.7

46 Chapter Topics in Analtic Geometr. 0. A 0, B, C cot A C B tan tan Solve for in terms of b completing the square: ± cot A C B Solve for in terms of : Graph 0 0 ± and.7 Graph ± 00 7 and 0. A, B, C cot A C B tan tan 7.. CONTINUED Houghton Mifflin Compan. All rights reserved.

47 Section. Rotation and Sstems of Quadratic Equations 7 0. CONTINUED Solve for in terms of with the Quadratic Formula: Graph 0 a, b, c b ± b ac a ± ± and B AC The graph is a hperbola. 0 cot A C B Matches graph (e). 0 0 The graph is a line. Matches graph B AC A, B, C The graph is a hperbola. B AC Houghton Mifflin Compan. All rights reserved.. cot A C B Matches graph (f) B AC cot A C B Matches graph (d). The graph is an ellipse or circle.. The graph is an ellipse or circle. cot A C B Matches graph (a). The graph is a parabola. cot A C B Matches graph A, B, C B AC 0..7

48 Chapter Topics in Analtic Geometr (a) B AC 0 Parabola ± ± (a) B AC. 0 ± ± > 0 Hperbola 7 0 (a) B AC ± 7 ±0 Ellipse or circle 0. (a). (a) B AC. 0 0 ± 0 0 ± 0 0 < 0 Ellipse or circle B AC 0 0 Parabola 0 0 (a) B AC 0 ± 0 ± Hperbola 0 ± ±00 0 Houghton Mifflin Compan. All rights reserved. 7

49 Section. Rotation and Sstems of Quadratic Equations. 0 (a) B AC 0 0 ±7 Parabola ±. (a) 0 ± B AC ± < 0 Ellipse or circle. 0. ± Two intersecting lines Point at, 0 (, ) ± ± Houghton Mifflin Compan. All rights reserved. ± Two parallel lines Two lines ± ±

50 0 Chapter Topics in Analtic Geometr Adding: 0 0, For, ±. is impossible. Solutions:,,, 7 0 ± 0 0 ± 0 No solution For : Solution:, or When : When : Points of intersection:,,, , 0 0 For 0: For : Houghton Mifflin Compan. All rights reserved. 0 0 Solutions: 0,,,

51 Section. Rotation and Sstems of Quadratic Equations or When 0: Point of intersection:, 0 When : 0 0 No real solution. 0 0 Houghton Mifflin Compan. All rights reserved.. When : 00 0 No real solution The point of intersection is 0,. In standard form the equations are: When : or ± When : or When : Points of intersection:,,,,,

52 Chapter Topics in Analtic Geometr ± Two solutions:,,, Solutions: 0,,, From Equation : In Equation : 0, impossible 7 0 One solution:, No solution Solution:, ( ) Note: has no real solution. When 0: When : The points of intersection are 0,,, 0. Houghton Mifflin Compan. All rights reserved. 0 or

53 Section. Rotation and Sstems of Quadratic Equations When 0 : When 0 : Points of intersection: ± , 0 0, 0,. True. B AC k. False. See Eample. However, A C A C. If k <, then B AC > 0.. g Asmptotes:, 0 Intercepts: 0, (0, ). f Intercept: 0, 0 Asmptotes:, (0, 0) Houghton Mifflin Compan. All rights reserved. 7. ht t t t t Slant asmptote: Vertical asmptote: Intercept: 0, 0 t t (0, 0) t. gs s Intercept: 0, Asmptotes: s ±, 0 ( 0, ) s

54 Chapter Topics in Analtic Geometr. (a). (a) AB BA 0 A BA A 0 does not eist. AB (a) AB 0 BA A (a) AB 7 BA A f. 0 f 0 0. g. g 7. ht t. ht t. 0 0 f t t f t t Houghton Mifflin Compan. All rights reserved.

55 Section. Parametric Equations 7. Area ab sin C 7. sin 0. Area ac sin B sin s 0 7. Area ss as bs c 7.0 s 7 Area ss as bs c 0. Section. Parametric Equations If f and g are continuous functions of t on an interval I, then the set of ordered pairs ft, gt is a plane curve C. The equations ft and gt are parametric equations for C and t is the parameter. You should be able to graph plane curves with our graphing utilit. To eliminate the parameter: Solve for t in one equation and substitute into the second equation. You should be able to find the parametric equations for a graph. Vocabular Check. plane curve, parametric equations, parameter. orientation. eliminating, parameter. t. t. t t t t t Houghton Mifflin Compan. All rights reserved.., line Matches. t t t Matches (a).. Parabola opening to the right Matches (d). ln t t e. t e Matches (f)., parabola, 0 Matches. t t e t e Eponential curve on 0 Matches (e).

56 Chapter Topics in Analtic Geometr 7. t, t (a) t Graph b hand. Note: 0 (d) t, Parabola In part, 0.. cos, sin (a) (d) cos sin, parabola The graph is an entire parabola rather than just the right portion.. The graph opens upward, contains, 0, and is oriented left to right. Matches.. t, t. 0. The orientation of the graph is clockwise and the center is,. Matches. t, t or 0 Houghton Mifflin Compan. All rights reserved.

58 Chapter Topics in Analtic Geometr. cos, sin. cos, sin cos, sin cos sin, ellipse cos sin, ellipse. e t et. e t e t, > 0, > 0 e t e t e t, > 0;, > ln t ln t ln ln 7 t t. ln t e t t e t e e Houghton Mifflin Compan. All rights reserved.

59 Section. Parametric Equations 7. cos, sin. cos, sin 0. sec, tan 0. sec tan. t e t lnt 0.t 0 0. B eliminating the parameters in (a) (d), we get. The differ from each other in restricted domain and in orientation. (a) Domain: < < Orientation: Left to right Domain: 0 < < Orientation: Right to left Domain: Orientation: Depends on (d) Domain: 0 < < Orientation: Left to right. Each curve represents a portion of the line 0. (a) t, 0 t, < < Houghton Mifflin Compan. All rights reserved. t, Orientation: Left to right t, < < t Orientation: Left to right (d) t Orientation: Left to right t, 0 t Orientation: Left to right for t 0 Right to left for t > 0

60 0 Chapter Topics in Analtic Geometr. t. h r cos k r sin h r cos, k r cos sin sin h k r r h k r 7. h a cos. h a sec k b sin k b tan h a cos, k b sin h a sec, k b tan h a k b sec tan h k a b. t t t 0. t t 7t h r cos cos k r sin sin. a, c, and b a c.. a, c, and b c a. The center is 0, 0, so h 0 and k 0. The center is 0, 0, so h 0 and k 0. cos sin so cos and, sin. This solution is not unique. sec tan tan., so sec and.... Answers will var. t, t t, t Sample answers: t, t t, t cot, sin Answers will var. t, 7t t, t Sample answers: t, t t, t 0. t t,. t t Sample answers: t, t t, t Sample answers: t, t t t t, t Houghton Mifflin Compan. All rights reserved.

61 Section. Parametric Equations. Matches.. Matches.. Matches (d).. Matches (a).. v 0 cos t, h v 0 sin t t (a) 00 mileshour.7 cos t 00 mihr 0 ftmi 00 sechr.7 ftsec.7 cos t.0t.7 sin t t 7.t t 0.7 sin t t.7 cos t.7t sin t t 0.0t t (d) Yes, it is a home run because > 0 when 00.. is the minimum angle It is not a home run because < 0 when 00.. (a) v 0 cos t 0 cos 0t h v 0 sin t t. 0 sin 0t t The horizontal distance is approimatel. feet. (d) You could use the Quadratic Formula to find the zeros of t 0 sin 0t.. The larger zero,., gives. feet The maimum height is approimatel 7. feet, when t.0 seconds. Houghton Mifflin Compan. All rights reserved. 7. (a) cos v 0 t If the ball is caught at time t, then: 7 sin v 0 t t 0 cos v 0 t 0 0 Maimum height feet (d) From part, t.0 seconds. 0 7 sin v 0 t t. v 0 t 0 cos sin 0 cos t t 0 tan t.0 seconds v ftsec t cos

62 Chapter Topics in Analtic Geometr. (a) v 0 cos t cos 0t h v 0 sin t t sin 0t t The maimum height is approimatel. feet when t.0 seconds. The horizontal distance is approimatel. feet. (d) You could solve the equation sin 0t t 0 for t.0. Then,. feet.. True t first set t t second set t t 0. False. The graph of t, t represents the portion of the line in the first quadrant.. False. For eample, t and t does not represent as a function of.. False. The equations represent a line.. Sample answer: cos sin. The graph is the same, but the orientation is reversed.. f Smmetric about the -ais Even function f. f, 0 No smmetr Neither even nor odd 7. e e ; e e No smmetr Neither even nor odd. No smmetr Neither even nor odd Houghton Mifflin Compan. All rights reserved.

Lesson 9.1 Using the Distance Formula

Lesson 9.1 Using the Distance Formula Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)