homework, page 673
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Homework, Page 673. Using the point P ( x, y ) and the rotation information, find the coordinates of P in the rotated x’y’ coordinate system. 33. Homework, Page 673. - PowerPoint PPT PresentationTRANSCRIPT
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1
Homework, Page 673Using the point P(x, y) and the rotation information, find the coordinates of P in the rotated x’y’ coordinate system.
33. , 2,5 , 4P x y
, 2,5 , 4P x y
cos sinx x y 2 22 5
2 2
2 2 5 2
2 2
3 2
2
cos siny y x 2 25 2
2 2
5 2 2 2
2 2
7 2
2
3 2 7 2, ,
2 2P x y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 2
Homework, Page 673Identify the type of conic, and rotate the coordinate system to eliminate the xy-term. Write and graph the transformed equation.
37. 8xy
8xy
x
y
cos sin sin cos 8x y x y 2 2 2 2cos sin cos sin sin cos 8x x y x y y
45 2 2 2 2cos sin cos sin 8x y x y
2 2 2 2 2 2 2 28
2 2 2 2 2 2x y x y
2 2 18
2x y
2 2
116 16
x y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 3
Homework, Page 673Identify the type of conic, solve for y, and graph the conic. Approximate the angle of rotation needed to eliminate the xy-term.
41. 2 216 20 9 40 0x xy y
2 4B AC 220 4 16 9 400 576 0 ellipse
2 29 20 16 40 0y x y x
2 220 20 4 9 16 40
2 9
x x xy
220 176 1440
18
x xy
cot 2A C
B 12 tan
B
A C
1 20
tan16 9
70.710
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4
Homework, Page 673Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
45. 2 29 6 7 5 0x xy y x y
2 4B AC 26 4 9 1 36 36 0 Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5
Homework, Page 673Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
49. 2 23 22 0x y y
2 4B AC 20 4 1 3 12 0 Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6
Homework, Page 67353. Find the center, vertices, and foci of the hyperbola in the original coordinate system.
2 9 0xy
2 2 2 2
2 2 2 2
2 2
2 2
2 9 0 2 cos sin sin cos 9
2 cos sin sin cos 9
2 cos sin cos sin sin cos 9
2 cos45 sin 45 cos 45 sin 45 9
2 2 2 22
2 2 2 2
xy x y x y
x y x y
x x y x y y
x y x y
x y x y
2 2
2 2
9
9 1 0,0 , 3 2,0 , 3,09 9
x yx y C F V
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7
Homework, Page 67353. Find the center, vertices, and foci of the hyperbola in the original coordinate system.
2 9 0xy
0,0 , 3 2,0 , 3,0
3 2 cos45 0sin 45 3 2 sin 45 0cos45 3, 3
3 2 3 23cos45 0sin 45 3sin 45 0cos45 ,
2 2
3 2 3 20,0 , 3, 3 , ,
2 2
C F V
C F V
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8
Homework, Page 67357.
True, because there is no xy term to cause a rotation.
2 2The graph of the equation 0 A and C not both zero
has a focal axis aligned with the coordinate axes. Justify your answer.
Ax Cy Dx Ey F
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9
Homework, Page 67361.
A. (1±4, –2)
B. (1±3, –2)
C. (4±1, 3)
D. (4±2, 3)
E. (1, –2±3)
2 2The vertices of 9 16 18 64 7 0 are:x y x y
2 2
2 2
2 2
2 2
2 2
2 2
2 2
9 16 18 64 71 0
9 18 16 64 71
9 2 16 4 71
9 2 1 16 4 4 71 9 64
9 1 16 2 144
9 1 16 2 144
144 144 144
1 21
16 91, 2 , 1 4, 2
x y x y
x x y y
x x y y
x x y y
x y
x y
x y
C F
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
8.5
Polar Equations of Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11
Quick Review
2
2 2
2 2
1. Solve for . (4, ) ( , )
2. Solve for . (3, 5 /3)=( 3, ), 2 2
3. Find the focus and the directrix of the parabola.
12
Find the focus and the vertices of the conic.
4. 116 9
5. 9 16
r r
x y
x y
x y
1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12
Quick Review Solutions
2
1. Solve for . (4, ) ( , )
2. Solve for . (3, 5 /3)=( 3, ), 2 2
3. Find the focus and the directrix of the parabola.
12
4
4 / 3
(0,3
Find the focus and the vertices of the conic.
)
4
; 3
r r
x y y
2 2
2 2
. 1 16 9
5.
( 5,0); ( 4,0)
(0, 7); 1 (0, 4) 9 16
x y
x y
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13
What you’ll learn about
Eccentricity Revisited Writing Polar Equations for Conics Analyzing Polar Equations of Conics Orbits Revisited
… and whyYou will learn the approach to conics used by astronomers.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 14
Focus-Directrix Definition Conic Section
A conic section is the set of all points in a plane whose distances from a particular point (the focus) and a particular line (the directrix) in the plane have a constant ratio. (We assume that the focus does not lie on the directrix.)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15
Focus-Directrix Eccentricity RelationshipIf is a point of a conic section, is the conic's focus, and is the
point of the directrix closest to , then and ,
where is a constant and the eccentricity of the conic.
Moreo
P F D
PFP e PF e PD
PDe
ver, the conic is
a hyperbola if 1,
a parabola if 1,
an ellipse if 1.
e
e
e
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16
A Conic Section in the Polar Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17
Three Types of Conics for r = ke/(1+ecosθ)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18
Polar Equations for Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19
Example Writing Polar Equations of Conics
Given that the focus is at the pole, write a polar equation for the conic
with eccentricity 4/5 and directrix 3.x
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20
Example Identifying Conics from Their Polar Equations
Determine the eccentricity, the type of conic, and the directrix.
6
3 2cosr
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 21
Example Matching Graphs of Conics with Their Polar Equations
Match the polar equation with its graph and identify the viewing window.
9
5 3sinr
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22
Example Finding Polar Equations of Conics
Find a polar equation for the ellipse with a focus at the pole and
the given polar coordinates as the endpoints of the major axis.
1.5,0 and 1,
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23
Example Finding Polar Equations of Conics
Find a polar equation for the hyperbola with a focus at the pole and
the given polar coordinates as the endpoints of its transverse axis.
3,0 and 1.5,
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24
Homework
Homework #23 Review Section 8.5 Page 682, Exercises: 1 – 29(EOO) Quiz next time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25
Semimajor Axes and Eccentricities of the Planets
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26
Ellipse with Eccentricity e and Semimajor Axis a
21
1 cos
a er
e