Transcript Slide 1

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. P  x, y    2,5 ,   4
P  x, y    2,5 ,  
4
 2  2
2 2 5 2 3 2

x  x cos   y sin   2 





5
 

2
2
2
 2   2 
 2
 2 5 2 2 2 7 2



y  y cos   x sin   5 
   2  

2
2
2
 2 
 2 
3 2 7 2 


P x , y   
,

2
2


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 1
Homework, Page 673
Identify the type of conic, and rotate the coordinate system to eliminate the xyterm. Write and graph the transformed equation.
37. xy  8
xy  8   x cos   y sin   x sin   y cos    8
x2 cos sin   xy cos2   xy sin 2   y2 sin  cos  8
  45  x2  y2 cos sin   xy cos2   sin 2   8





 2 2
2 2
2 2




x y
xy 

8
2 2
2
2
2
2


2
2


x
y
1

1
x2  y 2  8 
16
16
2
2
2




y







       

x
















Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 2
Homework, Page 673
Identify the type of conic, solve for y, and graph the conic. Approximate the
angle of rotation needed to eliminate the xy-term.
41. 16 x2  20 xy  9 y 2  40  0
B  4 AC   20   4 16  9   400  576  0  ellipse
2
2


9 y 2   20 x  y  16 x 2  40  0  y 
  20 x  
 20 x   4  9  16 x 2  40 
2 9
2
20 x  176 x 2  1440
y
18
AC
B
20
 70.710
cot 2 
 2  tan 1
 tan 1
B
AC
16  9
  
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 3
Homework, Page 673
Use the discriminant to decide whether the equation represents a parabola, an
ellipse, or a hyperbola.
45. 9 x2  6 xy  y 2  7 x  5 y  0
B  4 AC   6   4  9 1  36  36  0 Parabola
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 4
Homework, Page 673
Use the discriminant to decide whether the equation represents a parabola, an
ellipse, or a hyperbola.
49.
x2  3 y 2  y  22  0
B 2  4 AC   0 2  4 1 3  12  0 Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 5
Homework,
Page
673
Find the center, vertices, and foci of the hyperbola 2 xy  9  0 in the
53.
original coordinate system.
2 xy  9  0  2  x cos   y sin   x sin  y cos    9
2  x cos   y sin   x sin  y cos    9


2 x2 cos  sin   xy cos 2   xy sin 2   y2 sin  cos   9




2 x2  y2 cos 45 sin 45  xy cos 2 45  sin 2 45  9

2  x 2  y  2


2
2






2 2
2
2 
 xy  
 
  9
 2   2  
2 2


2
2


x
y
x 2  y  2  9 

 1  C   0,0  , F  3 2,0 ,V   3,0 
9
9



Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide 8- 6
Homework,
Page
673
Find the center, vertices, and foci of the hyperbola 2 xy  9  0 in the
53.
original coordinate system.


C   0,0  , F  3 2,0 ,V   3,0 
3


2 cos 45  0sin 45 3 2 sin 45  0cos 45   3, 3 
 3 2 3 2
  3cos 45  0sin 45 3sin 45  0cos 45      2 ,  2 


 3 2 3 2
C  0,0  , F  3, 3 ,V  
,

2
2


Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 7
Homework, Page 673
57. The graph of the equation Ax 2  Cy 2  Dx  Ey  F  0 A and C not both zero


has a focal axis aligned with the coordinate axes. Justify your answer.
True, because there is no xy term to cause a rotation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 8
61.
A.
B.
C.
D.
E.
Homework, Page 673
The vertices of 9 x2  16 y 2  18x  64 y  7  0 are:
(1±4, –2)
(1±3, –2)
(4±1, 3)
(4±2, 3)
(1, –2±3)
9 x 2  16 y 2  18 x  64 y  71  0
9 x 2  18 x  16 y 2  64 y  71

9 x



9 x 2  2 x  16 y 2  4 y  71
2



 2 x  1  16 y 2  4 y  4  71  9  64
9  x  1  16  y  2   144
2
9  x  1
144
 x  1
2
2
2

16  y  2 
2
144
 y  2


144
144
2
1
16
9
C 1, 2  , F 1  4, 2 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 9
8.5
Polar Equations of Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Quick Review
1. Solve for r. (4,  )  ( r ,    )
2. Solve for  . (3,  5 /3)=(  3,  ),  2    2
3. Find the focus and the directrix of the parabola.
x  12 y
Find the focus and the vertices of the conic.
2
2
2
2
2
x
y
4.
 1
16 9
x
y
5.

1
9 16
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 11
Quick Review Solutions
1. Solve for r. (4,  )  ( r ,    )  4
2. Solve for  . (3,  5 /3)=(  3,  ),  2    2 4 / 3
3. Find the focus and the directrix of the parabola.
x  12 y (0,3); y  3
Find the focus and the vertices of the conic.
2
2
2
2
2
x
y
4.
  1 (5, 0); (  4,0)
16 9
x
y
5.

 1 (0,  7); (0,  4)
9 16
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 12
What you’ll learn about




Eccentricity Revisited
Writing Polar Equations for Conics
Analyzing Polar Equations of Conics
Orbits Revisited
… and why
You will learn the approach to conics used by
astronomers.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 13
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- 14
Focus-Directrix Eccentricity Relationship
If P is a point of a conic section, F is the conic's focus, and D is the
PF
point of the directrix closest to P, then e 
and PF  e  PD,
PD
where e is a constant and the eccentricity of the conic.
Moreover, the conic is
a hyperbola if e  1,
a parabola if e  1,
an ellipse if e  1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 15
A Conic Section in the Polar Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 16
Three Types of Conics for r = ke/(1+ecosθ)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 17
Polar Equations for Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 18
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 x  3.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 19
Example Identifying Conics from Their
Polar Equations
Determine the eccentricity, the type of conic, and the directrix.
r
6
3  2cos 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 20
Example Matching Graphs of Conics
with Their Polar Equations
Match the polar equation with its graph and identify the viewing window.
r
9
5  3sin 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 21
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- 22
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- 23
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- 24
Semimajor Axes and Eccentricities of the
Planets
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 25
Ellipse with Eccentricity e and Semimajor
Axis a
r

a 1 e
2

1  e cos
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 26