Transcript ch.7 active learning

Active Learning Lecture Slides
For use with Classroom Response Systems
Chapter 7
Analytic
Geometry
© 2009 Pearson Education, Inc.
All rights reserved.
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 1
Find the equation of the parabola with focus at
(3, 0) and vertex at (0, 0).
a.
x 2  3y
b.
x  12y
c.
y 2  3x
d.
y  12x
2
2
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 2
Find the equation of the parabola with focus at
(3, 0) and vertex at (0, 0).
a.
x 2  3y
b.
x  12y
c.
y 2  3x
d.
y  12x
2
2
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 3
Find the vertex, focus, and directrix of
2
y  1   x  3.
a.
V: (3, 1)
F: (2.75, 1)
D: x = 3.25
b.
V: (–1, –3)
F: (–1.25, –3)
D: x = 2.75
c.
V: (3, 1)
F: (3, 0.75)
D: y = 1.25
d.
V: (–3, –1)
F: (–3, –1.25)
D: y = –0.75
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 4
Find the vertex, focus, and directrix of
2
y  1   x  3.
a.
V: (3, 1)
F: (2.75, 1)
D: x = 3.25
b.
V: (–1, –3)
F: (–1.25, –3)
D: x = 2.75
c.
V: (3, 1)
F: (3, 0.75)
D: y = 1.25
d.
V: (–3, –1)
F: (–3, –1.25)
D: y = –0.75
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 5
A reflecting telescope contains a mirror shaped
like a paraboloid of revolution. If the mirror is 24
inches across at its opening and is 2 feet deep,
where will the light be concentrated?
a. 18 in. from the vertex
b. 1.5 in. from the vertex
c. 0.2 in. from the vertex
d. 0.1 in. from the vertex
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 6
A reflecting telescope contains a mirror shaped
like a paraboloid of revolution. If the mirror is 24
inches across at its opening and is 2 feet deep,
where will the light be concentrated?
a. 18 in. from the vertex
b. 1.5 in. from the vertex
c. 0.2 in. from the vertex
d. 0.1 in. from the vertex
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 7
Find an equation for the ellipse with center at
(0, 0), focus at (2, 0) and vertex at (6, 0).
2
2
a.
x
y

1
4 36
c.
x 2 y2

1
4 32
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2
2
b.
x
y

1
32 36
d.
x 2 y2

1
36 32
Slide 7 - 8
Find an equation for the ellipse with center at
(0, 0), focus at (2, 0) and vertex at (6, 0).
2
2
a.
x
y

1
4 36
c.
x 2 y2

1
4 32
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2
2
b.
x
y

1
32 36
d.
x 2 y2

1
36 32
Slide 7 - 9
Find the center, foci, and vertices of the ellipse
2
2
36 x  3  9 y  1  324.
a. C: (–3, 1) V: (–9, 1), (3, 1)
F : 3  3 3,1 , 3  3 3,1



b. C: (1, –3) V: (–9, 1), (3, 1)
F : 1 3 3, 3 , 1 3 3, 3



c. C: (–3, 1) V: (6, 1), (–6, 1)
F : 3 3,1 , 3 3,1



d. C: (–3, 1) V: (6, 1), (–6, 1)
F : 3  3 3,3 , 3  3 3, –3

Copyright © 2009 Pearson Education, Inc.


Slide 7 - 10
Find the center, foci, and vertices of the ellipse
2
2
36 x  3  9 y  1  324.
a. C: (–3, 1) V: (–9, 1), (3, 1)
F : 3  3 3,1 , 3  3 3,1



b. C: (1, –3) V: (–9, 1), (3, 1)
F : 1 3 3, 3 , 1 3 3, 3



c. C: (–3, 1) V: (6, 1), (–6, 1)
F : 3 3,1 , 3 3,1



d. C: (–3, 1) V: (6, 1), (–6, 1)
F : 3  3 3,3 , 3  3 3, –3

Copyright © 2009 Pearson Education, Inc.


Slide 7 - 11
Graph 9 x  1  4 y  2   36.
2
2
a.
b.
c.
d.
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Slide 7 - 12
Graph 9 x  1  4 y  2   36.
2
2
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 13
A bridge is built in the shape of a semielliptical
arch. It has a span of 110 feet. The height of the
arch 29 feet from the center is to be 6 feet. Find
the height of the arch at its center.
a. 6.22 ft
b. 7.06 ft
c. 29.17 ft
d. 11.38 ft
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 14
A bridge is built in the shape of a semielliptical
arch. It has a span of 110 feet. The height of the
arch 29 feet from the center is to be 6 feet. Find
the height of the arch at its center.
a. 6.22 ft
b. 7.06 ft
c. 29.17 ft
d. 11.38 ft
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 15
Find an equation for the hyperbola with
5
vertices at (0, ±10) and asymptote the line y  x.
6
2
a.
2
y
x

1
100 36
2
2
y
x
c.

1
100 144
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2
2
b.
y
x

1
144 100
d.
y2 x 2

1
36 25
Slide 7 - 16
Find an equation for the hyperbola with
5
vertices at (0, ±10) and asymptote the line y  x.
6
2
a.
2
y
x

1
100 36
2
2
y
x
c.

1
100 144
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2
2
b.
y
x

1
144 100
d.
y2 x 2

1
36 25
Slide 7 - 17
Find the asymptotes
of the hyperbola
 x  2
25
2
y  1


2
16
a.
4
4
y  1  x  2 ; y  1   x  2 
5
5
b.
5
5
y  1  x  2 ; y  1   x  2 
4
4
c.
4
4
y  x  2 ; y   x  2 
5
5
d.
4
4
y  2  x  1; y  2   x  1
5
5
Copyright © 2009 Pearson Education, Inc.
 1.
Slide 7 - 18
Find the asymptotes
of the hyperbola
 x  2
25
2
y  1


2
16
a.
4
4
y  1  x  2 ; y  1   x  2 
5
5
b.
5
5
y  1  x  2 ; y  1   x  2 
4
4
c.
4
4
y  x  2 ; y   x  2 
5
5
d.
4
4
y  2  x  1; y  2   x  1
5
5
Copyright © 2009 Pearson Education, Inc.
 1.
Slide 7 - 19
Find an equation for the hyperbola with center at
(7, 8), focus at (3, 8), and vertex at (6, 8).
a. x  7 
y  8


c. x  8 
y  7


2
2
2
15
Copyright © 2009 Pearson Education, Inc.
 1 b.
2
15
x  7 
2
15
x  8 
2
 1 d.
15
 y  8   1
2
 y  7   1
2
Slide 7 - 20
Find an equation for the hyperbola with center at
(7, 8), focus at (3, 8), and vertex at (6, 8).
a. x  7 
y  8


c. x  8 
y  7


2
2
2
15
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 1 b.
2
15
x  7 
2
15
x  8 
2
 1 d.
15
 y  8   1
2
 y  7   1
2
Slide 7 - 21
y  1
2
Graph
4
x  2


2
25
a.
b.
c.
d.
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 1.
Slide 7 - 22
y  1
2
Graph
4
x  2


2
25
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
 1.
Slide 7 - 23
Two recording devices are set 3000 feet apart, with
the device at point A to the west of the device at
point B. At a point on a line between the devices,
300 feet from point B, s small amount of explosive
is detonated. The recording devices record the time
the second reaches each one. How far directly north
of site B should a second explosion be done so that
the measured time difference recorded by the
devices is the same as that for the first detonation?
a. 1440.7 ft
b.
4409.08 ft
c. 1469.69 ft
d.
675 ft
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 24
Two recording devices are set 3000 feet apart, with
the device at point A to the west of the device at
point B. At a point on a line between the devices,
300 feet from point B, s small amount of explosive
is detonated. The recording devices record the time
the second reaches each one. How far directly north
of site B should a second explosion be done so that
the measured time difference recorded by the
devices is the same as that for the first detonation?
a. 1440.7 ft
b.
4409.08 ft
c. 1469.69 ft
d.
675 ft
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 25
Identify the equation 3x  3y  3x  2y  3  0.
2
2
a. parabola
b. ellipse
c. hyperbola
d. not a conic
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 26
Identify the equation 3x  3y  3x  2y  3  0.
2
2
a. parabola
b. ellipse
c. hyperbola
d. not a conic
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 27
Determine the rotation formulas to use so that
the new equation contains no xy-term.
8x  6 3xy  2y  41  0
2
a. x 
b. x 
c. x 
d. x 




1
3x   y 
2
1
3x   y 
2
1
x   3y 
2
1
2 x   y
2
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2
;
;
;
;








1
y  x   3y 
2
1
y  x   3y 
2
1
y
3x   y 
2
1
y  x   2 y
2
Slide 7 - 28
Determine the rotation formulas to use so that
the new equation contains no xy-term.
8x  6 3xy  2y  41  0
2
a. x 
b. x 
c. x 
d. x 




1
3x   y 
2
1
3x   y 
2
1
x   3y 
2
1
2 x   y
2
Copyright © 2009 Pearson Education, Inc.
2
;
;
;
;








1
y  x   3y 
2
1
y  x   3y 
2
1
y
3x   y 
2
1
y  x   2 y
2
Slide 7 - 29
Rotate the axes so that the new equation
contains no xy-term. Give the angle of rotation.
24xy  7y  36  0
2
a.
  53.1º
2
2
y
4x

1
4
9
c.   36.9º
2
2
y
x

1
4 16
Copyright © 2009 Pearson Education, Inc.
b.   36.9º
2
2
y
4x

1
4
9
d.   36.9º
4 y2 x 2

1
9
4
Slide 7 - 30
Rotate the axes so that the new equation
contains no xy-term. Give the angle of rotation.
24xy  7y  36  0
2
a.
  53.1º
2
2
y
4x

1
4
9
c.   36.9º
2
2
y
x

1
4 16
Copyright © 2009 Pearson Education, Inc.
b.   36.9º
2
2
y
4x

1
4
9
d.   36.9º
4 y2 x 2

1
9
4
Slide 7 - 31
Identify the equation
2x 2  6xy  9y 2  3x  3y  8  0.
a. parabola
b. ellipse
c. hyperbola
d. not a conic
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 32
Identify the equation
2x 2  6xy  9y 2  3x  3y  8  0.
a. parabola
b. ellipse
c. hyperbola
d. not a conic
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 33
Identify the conic that the polar equation
represents and give the position of the directrix.
9
r
1  3cos 
a. hyperbola; directrix perpendicular to the
polar axis 3 left of the pole
b. hyperbola; directrix perpendicular to the
polar axis 3 right of the pole
c. ellipse; directrix perpendicular to the polar
axis 3 left of the pole
d. ellipse; directrix perpendicular to the polar
axis 3 right of the pole
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 34
Identify the conic that the polar equation
represents and give the position of the directrix.
9
r
1  3cos 
a. hyperbola; directrix perpendicular to the
polar axis 3 left of the pole
b. hyperbola; directrix perpendicular to the
polar axis 3 right of the pole
c. ellipse; directrix perpendicular to the polar
axis 3 left of the pole
d. ellipse; directrix perpendicular to the polar
axis 3 right of the pole
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 35
8
Convert r 
to a rectangular equation.
2  sin 
a.
3x 2  4y 2  16x  64  0
b.
4x  3y  16x  64  0
c.
5x 2  4y 2  16x  64  0
d.
4x  4y  16x  64  0
2
2
2
2
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Slide 7 - 36
8
Convert r 
to a rectangular equation.
2  sin 
a.
3x 2  4y 2  16x  64  0
b.
4x  3y  16x  64  0
c.
5x 2  4y 2  16x  64  0
d.
4x  4y  16x  64  0
2
2
2
2
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 37
Graph the curve whose parametric equations
are x = 2t – 1, y = t2 + 2; –4 ≤ t ≤ 4.
a.
b.
c.
d.
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Slide 7 - 38
Graph the curve whose parametric equations
are x = 2t – 1, y = t2 + 2; –4 ≤ t ≤ 4.
a.
b.
c.
d.
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Slide 7 - 39
Use a graphing utility to graph the curve
whose parametric equations are
x  3cos t , y  2sin t;  4  t  2
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 40
Use a graphing utility to graph the curve
whose parametric equations are
x  3cos t , y  2sin t;  4  t  2
a.
b.
c.
d.
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 41
Find a rectangular equation for the plane curve
defined by x = 9sint, y = 9cost; 0 ≤ t ≤ 2π.
a.
x  y  81;  9  x  9
b.
x 2  y 2  81;  •  x  •
c.
y  x 2  81;  •  x  •
d.
y  x  9;  2  x  2
2
2
2
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 42
Find a rectangular equation for the plane curve
defined by x = 9sint, y = 9cost; 0 ≤ t ≤ 2π.
a.
x  y  81;  9  x  9
b.
x 2  y 2  81;  •  x  •
c.
y  x 2  81;  •  x  •
d.
y  x  9;  2  x  2
2
2
2
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 43
A baseball player hit a baseball with an initial speed
of 190 feet per second at an angle of 40º to the
horizontal. The ball was a hit at a height of 5 feet off
the ground. Find the parametric equations that
describe the motion of the ball as a function of time.
a. x  145.54t;
y  16t 2  122.17t  5
1105.231 feet
c. x  145.54t;
y  16t 2  122.17t  5
1117.165 feet
Copyright © 2009 Pearson Education, Inc.
b. x  145.54t;
y  16t 2  122.17t  5
2234.476 feet
d. x  145.54t;
y  16t 2  122.17t  5
1880.513 feet
Slide 7 - 44
A baseball player hit a baseball with an initial speed
of 190 feet per second at an angle of 40º to the
horizontal. The ball was a hit at a height of 5 feet off
the ground. Find the parametric equations that
describe the motion of the ball as a function of time.
a. x  145.54t;
y  16t 2  122.17t  5
1105.231 feet
c. x  145.54t;
y  16t 2  122.17t  5
1117.165 feet
Copyright © 2009 Pearson Education, Inc.
b. x  145.54t;
y  16t 2  122.17t  5
2234.476 feet
d. x  145.54t;
y  16t 2  122.17t  5
1880.513 feet
Slide 7 - 45
Find parametric equations for y = 9x + 5.
a.
t
x ; yt5
9
b. x  9t; y  t  5
c.
t
x  t; y   5
9
t
d. x  ; y  t  5
9
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 46
Find parametric equations for y = 9x + 5.
a.
t
x ; yt5
9
b. x  9t; y  t  5
c.
t
x  t; y   5
9
t
d. x  ; y  t  5
9
Copyright © 2009 Pearson Education, Inc.
Slide 7 - 47
Find parametric equations for
0 ≤ t ≤ 2 that define the curve.




a. x  7 sin  t  1 , y  8 cos  t  1
2

2





b. x  7 sin  t  1 , y  8 cos  t  1
2

2





c. x  8 sin  t  1 , y  7 cos  t  1
2

2





d. x  8 sin  t  1 , y  7 cos  t  1
2

2

Copyright © 2009 Pearson Education, Inc.
Slide 7 - 48
Find parametric equations for
0 ≤ t ≤ 2 that define the curve.




a. x  7 sin  t  1 , y  8 cos  t  1
2

2





b. x  7 sin  t  1 , y  8 cos  t  1
2

2





c. x  8 sin  t  1 , y  7 cos  t  1
2

2





d. x  8 sin  t  1 , y  7 cos  t  1
2

2

Copyright © 2009 Pearson Education, Inc.
Slide 7 - 49