Transcript Slide 1

Homework, Page 682
Find a polar equation for the conic with a focus at the pole and the
given eccentricity and directrix. Identify the conic and graph it.
1. e  1, x  2
e  1, x  2  k  2
ke
 2 1  r  2
r

1  e cos  1  1 cos
1  cos
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 1
Homework, Page 682
Find a polar equation for the conic with a focus at the pole and the
given eccentricity and directrix. Identify the conic and graph it.
5. e  7 3 , y  1
e  7 , y  1  k  1
3
1 7 3
ke
3

r
1  e sin 
1  7 cos 3
3
 
 
 r
7
3  7sin 
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Slide 8- 2
Homework, Page 682
Determine the eccentricity, type of conic and directrix.
r
9.
5
2  2sin 
ke
1  e sin 
5
r
2  2sin 
r
k  2.5 , e  1
1
2  2.5
1
1  1sin 
2
parabola , opening up , vertex at  0, 1.25  , directrix: y  2.5
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 3
Homework, Page 682
Determine the eccentricity, type of conic and directrix.
13. r 
6
5  2cos 
ke
1  e cos 
6
r
5  2cos 
r
1
6
5
5
1
1  2 cos
5
5
6
e  2 ,k  5  3
5
2
5
ellipse , directrix: x  3
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Slide 8- 4
Homework, Page 682
Match the polar equation with its graph and identify the viewing window.
17.
r
5
2  2sin 
ke
1  e sin 
5
r
2  2sin 
r
k  2.5 , e  1
1
2  2.5
1
1  1sin 
2
parabola , opening up
vertex at  0, 1.25 
directrix: y  2.5
Graph (f), lower right
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 5
Homework, Page 682
Find a polar equation for an ellipse with a focus at the pole and the
given polar coordinates as the endpoints of its major axis..
21. 1.5,0  and  6,  
y

M
1.5   6 

 2.25
2
a  1.5   2.25   3.75  15 4


c  0   2.25   2.25  9 4
9
c
e  4 9 3
a 15  
4 15 5
3
k
ke
5
3k
5

r

1  e cos  1  3 cos 5 5  3cos 
5
12
3
3k
3
3k
3 8

 

 k k  4 r 
5  3cos 
2 5  3cos 0
2 53 2 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


x










Slide 8- 6

Homework, Page 682
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..
25.  3,0  and  15,  
y

15  3  9
M
2
c 9 3

e

 
,
a

15

9

6
c9
a 6 2
3
3
k
k
ke
2 
2
3
r
3
2 3
1  e cos 
1  cos 0
 1
2
2 2







  

x








       








3
3
5
2
35
2
15
3 2 k 
 5 r 

r

5
3
3
2  3cos 
1  cos 2
2
2
k
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 7
Homework, Page 682
Find a polar equation for the conic with a focus at the pole.
29.
y
(3, pi)
(0.75, 0)
x
 0.75   3   9 
M 
,0     ,0 
2

  8 
3  9
a       15  c  0    9   9


4  8
8
 8 8
3
k
9
5
9 3
ke
3k
5
c
8


 r 

e 
15 5
1  e cos 1  3 5 cos  5 5  3cos 
a 15
8
6
3k
3
3k
3 8


k 
 2 r 
5  3cos 
5  3 1
4 5  3cos 0
4 3
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 8
What you’ll learn about


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- 9
Example Analyzing Polar Equations of
Conics
Graph the conic and find the values of e, a, b, and c.
16
r

34.
5  3cos 
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 10
Example Analyzing Polar Equations of
Conics
Determine a Cartesian equivalent for the given polar equation.
6
r

38.
1  2cos 
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Slide 8- 11
Example, Writing Cartesian Equations
From Polar Equations
Use the fact that k = 2p is twice the focal length and half the focal width to
determine the Cartesian equation of the parabola whose polar equation is given.
12
r

40.
3  3cos 
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Slide 8- 12
Semimajor Axes and Eccentricities of the
Planets
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Slide 8- 13
Ellipse with Eccentricity e and Semimajor
Axis a
r

a 1  e2

1  e cos
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 14
Example Analyzing Orbits of Planets
42.
The orbit of the planet Uranus has a semimajor axis of
19.18 AU and an orbital eccentricity of 0.0461. Compute its
perihelion and aphelion distances.
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Slide 8- 15
Homework



Homework Assignment #24
Read Section 8.6
Page 682, Exercises: 33 – 49(Odd), skip 43
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Slide 8- 16
8.6
Three-Dimensional Cartesian
Coordinate System
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Quick Review
Let P( x, y ) and Q(3, 2) be points in the xy -plane.
1. Compute the distance between P and Q.
2. Find the midpoint of the line segment PQ.
3. If P is 5 units from Q, describe the position of P.
Let v  4,5 be a vector in the xy - plane.
4. Find the maginitude of v.
5. Find a unit vector in the direction of v.
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Slide 8- 18
Quick Review Solutions
Let P( x, y ) and Q(3, 2) be points in the xy -plane.
1. Compute the distance between P and Q.
 x  3   y  2 
2
2
 x3 y2
2. Find the midpoint of the line segment PQ. 
,

2 
 2
3. If P is 5 units from Q, describe the position of P.  x  3   y  2   25
2
2
Let v  4,5 be a vector in the xy - plane.
4. Find the maginitude of v. 41
5. Find a unit vector in the direction of v.
4
41
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,
5
41
Slide 8- 19
What you’ll learn about






Three-Dimensional Cartesian Coordinates
Distances and Midpoint Formula
Equation of a Sphere
Planes and Other Surfaces
Vectors in Space
Lines in Space
… and why
This is the analytic geometry of our physical world.
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Slide 8- 20
The Point P(x,y,z) in Cartesian Space
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Slide 8- 21
Features of the Three-Dimensional
Cartesian Coordinate System






The axes are labeled x, y, and z, and these three coordinate
axes form a right-handed coordinate frame.
The Cartesian coordinates of a point P in space are an
ordered triple, (x, y, z).
Pairs of axes determine the coordinate planes.
The coordinate planes are the xy-plane, the xz-plane, and the
yz-plane and they have equations z = 0, y = 0, and x = 0,
respectively.
The coordinate planes meet at the origin (0, 0, 0).
The coordinate planes divide space into eight regions called
octants. The first octant contains all points in space with
three positive coordinates.
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Slide 8- 22
The Coordinate Planes Divide Space into
Eight Octants
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Slide 8- 23
Distance Formula (Cartesian Space)
The distance d ( P, Q) between the points P( x1 , y1 , z1 ) and Q( x2 , y2 , z2 )
in space is d ( P, Q) 
 x1  x2    y1  y2    z1  z2 
2
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2
2
.
Slide 8- 24
Midpoint Formula (Cartesian Space)
The midpoint M of the line segment PQ with endpoints P( x1 , y1 , z1 )
 x1  x2 y1  y2 z1  z2 
and Q( x2 , y2 , z2 ) is M  
,
,
.
2
2 
 2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 25
Example Calculating a Distance and
Finding a Midpoint
Find the distance between the points P(1, 2,3) and Q(4,5,6), and
find the midpoint of the line segment PQ.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 26
Standard Equation of a Sphere
A point P( x, y, z ) is on the sphere with center (h, k , l ) and radius r
if and only if  x  h    y  k    z  l   r 2 .
2
2
2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 27
Drawing Lesson
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Slide 8- 28
Drawing Lesson (cont’d)
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Slide 8- 29
Example Finding the Standard Equation
of a Sphere
Find the standard equation of the sphere with center (1, 2,3)
and radius 4.
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Slide 8- 30
Equation for a Plane in Cartesian Space
Every plane can be written as Ax  By  Cz  D  0, where A, B, and C
are not all zero. Conversely, every first-degree equation in three
variables represents a plane in Cartesian space.
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Slide 8- 31
The Vector v = <v1,v2,v3>
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Slide 8- 32
Vector Relationships in Space
For vectors v  v1 , v2 , v3 and w  w1 , w2 , w3 ,
Equality : v = w if and only if v1  w1 , v2  w2 , v3  w3
Addition : v + w = v1  w1 , v2  w2 , v3  w3
Subtraction : v  w = v1  w1 , v2  w2 , v3  w3
Magnitude : v  v12  v2 2  v32
Dot Product : v  w =v1w1  v2 w2 +v3 w3
Unit Vector : u  v / v , v  0, is the unit vector in the direction of v.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 33
Equations for a Line in Space
If is a line through the point PO ( xO , yO , zO ) in the direction of a
nonzero vector v  a, b, c , then a point P( x, y, z ) is on if and only if
Vector form: r  rO  tv , where r  x, y, z and rO  xO , yO , zO ; or
Parametric form: x  xO  at , y  yO  bt , and z  zO  ct , where t is
a real number.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 34
Example Finding Equations for a Line
Using the standard unit vector i , j, and k , write a vector equation for
the line containing the points A(2,0,3) and B(4, 1,3).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 8- 35