slide 7 - 1 copyright © 2009 pearson education, inc. active learning lecture slides for use with...
TRANSCRIPT
Slide 7 - 1Copyright © 2009 Pearson Education, Inc.
Active Learning Lecture SlidesFor use with Classroom Response Systems
© 2009 Pearson Education, Inc.All rights reserved.
Chapter 7Analytic
Geometry
Slide 7 - 2Copyright © 2009 Pearson Education, Inc.
Find the equation of the parabola with focus at (3, 0) and vertex at (0, 0).
a.
b.
c.
d.
x2 3y
x2 12y
y2 3x
y2 12x
Slide 7 - 3Copyright © 2009 Pearson Education, Inc.
Find the equation of the parabola with focus at (3, 0) and vertex at (0, 0).
a.
b.
c.
d.
x2 3y
x2 12y
y2 3x
y2 12x
Slide 7 - 4Copyright © 2009 Pearson Education, Inc.
Find the vertex, focus, and directrix ofy 1 2 x 3 .
b.
c. d.
a. V: (3, 1)F: (2.75, 1)D: x = 3.25
V: (–1, –3)F: (–1.25, –3)D: x = 2.75
V: (3, 1)F: (3, 0.75)D: y = 1.25
V: (–3, –1)F: (–3, –1.25)D: y = –0.75
Slide 7 - 5Copyright © 2009 Pearson Education, Inc.
Find the vertex, focus, and directrix ofy 1 2 x 3 .
b.
c. d.
a. V: (3, 1)F: (2.75, 1)D: x = 3.25
V: (–1, –3)F: (–1.25, –3)D: x = 2.75
V: (3, 1)F: (3, 0.75)D: y = 1.25
V: (–3, –1)F: (–3, –1.25)D: y = –0.75
Slide 7 - 6Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 7Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 8Copyright © 2009 Pearson Education, Inc.
Find an equation for the ellipse with center at (0, 0), focus at (2, 0) and vertex at (6, 0).
a.
c. x2
36y2
321
x2
4y2
321
x2
32y2
361
x2
4y2
361 b.
d.
Slide 7 - 9Copyright © 2009 Pearson Education, Inc.
Find an equation for the ellipse with center at (0, 0), focus at (2, 0) and vertex at (6, 0).
a.
c. x2
36y2
321
x2
4y2
321
x2
32y2
361
x2
4y2
361 b.
d.
Slide 7 - 10Copyright © 2009 Pearson Education, Inc.
Find the center, foci, and vertices of the ellipse
a. C: (–3, 1) V: (–9, 1), (3, 1)
F : 3 3 3,1 , 3 3 3,1
36 x 3 2 9 y 1 2 324.
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
Slide 7 - 11Copyright © 2009 Pearson Education, Inc.
Find the center, foci, and vertices of the ellipse
a. C: (–3, 1) V: (–9, 1), (3, 1)
F : 3 3 3,1 , 3 3 3,1
36 x 3 2 9 y 1 2 324.
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
Slide 7 - 12Copyright © 2009 Pearson Education, Inc.
Graph
a.
9 x 1 2 4 y 2 2 36.
b.
c. d.
Slide 7 - 13Copyright © 2009 Pearson Education, Inc.
Graph
a.
9 x 1 2 4 y 2 2 36.
b.
c. d.
Slide 7 - 14Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 15Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 16Copyright © 2009 Pearson Education, Inc.
Find an equation for the hyperbola with vertices at (0, ±10) and asymptote the line
a.
c. y2
36x2
251
y2
100x2
1441
y2
144x2
1001
y2
100x2
361 b.
d.
y5
6x.
Slide 7 - 17Copyright © 2009 Pearson Education, Inc.
Find an equation for the hyperbola with vertices at (0, ±10) and asymptote the line
a.
c. y2
36x2
251
y2
100x2
1441
y2
144x2
1001
y2
100x2
361 b.
d.
y5
6x.
Slide 7 - 18Copyright © 2009 Pearson Education, Inc.
Find the asymptotes of the hyperbola
a. y 1 4
5x 2 ; y 1
4
5x 2
2 22 1
1.25 16
x y
b.
c.
d.
y 1 5
4x 2 ; y 1
5
4x 2
y4
5x 2 ; y
4
5x 2
y 2 4
5x 1 ; y 2
4
5x 1
Slide 7 - 19Copyright © 2009 Pearson Education, Inc.
Find the asymptotes of the hyperbola
a. y 1 4
5x 2 ; y 1
4
5x 2
2 22 1
1.25 16
x y
b.
c.
d.
y 1 5
4x 2 ; y 1
5
4x 2
y4
5x 2 ; y
4
5x 2
y 2 4
5x 1 ; y 2
4
5x 1
Slide 7 - 20Copyright © 2009 Pearson Education, Inc.
Find an equation for the hyperbola with center at (7, 8), focus at (3, 8), and vertex at (6, 8).
a.
c.x 8 2
15 y 7 2 1x 8 2
y 7 2
151
x 7 2
15 y 8 2 1x 7 2
y 8 2
151 b.
d.
Slide 7 - 21Copyright © 2009 Pearson Education, Inc.
Find an equation for the hyperbola with center at (7, 8), focus at (3, 8), and vertex at (6, 8).
a.
c.x 8 2
15 y 7 2 1x 8 2
y 7 2
151
x 7 2
15 y 8 2 1x 7 2
y 8 2
151 b.
d.
Slide 7 - 22Copyright © 2009 Pearson Education, Inc.
Graph
a. b.
c. d.
y 1 2
4x 2 2
251.
Slide 7 - 23Copyright © 2009 Pearson Education, Inc.
Graph
a. b.
c. d.
y 1 2
4x 2 2
251.
Slide 7 - 24Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 25Copyright © 2009 Pearson Education, Inc.
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
Slide 7 - 26Copyright © 2009 Pearson Education, Inc.
Identify the equation
a. parabola
b. ellipse
c. hyperbola
d. not a conic
3x2 3y2 3x 2y 3 0.
Slide 7 - 27Copyright © 2009 Pearson Education, Inc.
Identify the equation
a. parabola
b. ellipse
c. hyperbola
d. not a conic
3x2 3y2 3x 2y 3 0.
Slide 7 - 28Copyright © 2009 Pearson Education, Inc.
Determine the rotation formulas to use so that the new equation contains no xy-term.
a.
b.
c.
d.
x 1
23 x y ; y
1
2x 3 y
x 1
23 x y ; y
1
2x 3 y
x 1
2x 3 y ; y
1
23 x y
x 1
22 x y ; y
1
2x 2 y
8x2 6 3xy 2y2 41 0
Slide 7 - 29Copyright © 2009 Pearson Education, Inc.
Determine the rotation formulas to use so that the new equation contains no xy-term.
a.
b.
c.
d.
x 1
23 x y ; y
1
2x 3 y
x 1
23 x y ; y
1
2x 3 y
x 1
2x 3 y ; y
1
23 x y
x 1
22 x y ; y
1
2x 2 y
8x2 6 3xy 2y2 41 0
Slide 7 - 30Copyright © 2009 Pearson Education, Inc.
Rotate the axes so that the new equation contains no xy-term. Give the angle of rotation.
a.
c. 36.9º
4y2
9x2
41
36.9º
y2
4x2
161
36.9º
y2
4
4x2
91
53.1º
y2
4
4x2
91
b.
d.
24xy 7y2 36 0
Slide 7 - 31Copyright © 2009 Pearson Education, Inc.
Rotate the axes so that the new equation contains no xy-term. Give the angle of rotation.
a.
c. 36.9º
4y2
9x2
41
36.9º
y2
4x2
161
36.9º
y2
4
4x2
91
53.1º
y2
4
4x2
91
b.
d.
24xy 7y2 36 0
Slide 7 - 32Copyright © 2009 Pearson Education, Inc.
Identify the equation
a. parabola
b. ellipse
c. hyperbola
d. not a conic
2x2 6xy 9y2 3x 3y 8 0.
Slide 7 - 33Copyright © 2009 Pearson Education, Inc.
Identify the equation
a. parabola
b. ellipse
c. hyperbola
d. not a conic
2x2 6xy 9y2 3x 3y 8 0.
Slide 7 - 34Copyright © 2009 Pearson Education, Inc.
Identify the conic that the polar equation represents and give the position of the directrix.
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
r 9
1 3cos
Slide 7 - 35Copyright © 2009 Pearson Education, Inc.
Identify the conic that the polar equation represents and give the position of the directrix.
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
r 9
1 3cos
Slide 7 - 36Copyright © 2009 Pearson Education, Inc.
Convert
a.
b.
c.
d.
3x2 4y2 16x 64 0
4x2 3y2 16x 64 0
5x2 4y2 16x 64 0
4x2 4y2 16x 64 0
to a rectangular equation.8
2 sinr
Slide 7 - 37Copyright © 2009 Pearson Education, Inc.
Convert
a.
b.
c.
d.
3x2 4y2 16x 64 0
4x2 3y2 16x 64 0
5x2 4y2 16x 64 0
4x2 4y2 16x 64 0
to a rectangular equation.8
2 sinr
Slide 7 - 38Copyright © 2009 Pearson Education, Inc.
Graph the curve whose parametric equations are x = 2t – 1, y = t2 + 2; –4 ≤ t ≤ 4.
a. b.
c. d.
Slide 7 - 39Copyright © 2009 Pearson Education, Inc.
Graph the curve whose parametric equations are x = 2t – 1, y = t2 + 2; –4 ≤ t ≤ 4.
a. b.
c. d.
Slide 7 - 40Copyright © 2009 Pearson Education, Inc.
Use a graphing utility to graph the curve whose parametric equations are
a. b.
c. d.
3cos , 2sin ; 4 2x t y t t
Slide 7 - 41Copyright © 2009 Pearson Education, Inc.
Use a graphing utility to graph the curve whose parametric equations are
a. b.
c. d.
3cos , 2sin ; 4 2x t y t t
Slide 7 - 42Copyright © 2009 Pearson Education, Inc.
Find a rectangular equation for the plane curve defined by x = 9 sin t, y = 9 cos t; 0 ≤ t ≤ 2π.
a.
b.
c.
d.2 9; 2 2y x x
y x2 81; x
2 2 81; 9 9x y x
x2 y2 81; x
Slide 7 - 43Copyright © 2009 Pearson Education, Inc.
Find a rectangular equation for the plane curve defined by x = 9 sin t, y = 9 cos t; 0 ≤ t ≤ 2π.
a.
b.
c.
d.2 9; 2 2y x x
y x2 81; x
2 2 81; 9 9x y x
x2 y2 81; x
Slide 7 - 44Copyright © 2009 Pearson Education, Inc.
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.
c.
x 145.54t;
y 16t 2 122.17t 5
2234.476 feet
x 145.54t;
y 16t 2 122.17t 5
1105.231 feet
b.
d. x 145.54t;
y 16t 2 122.17t 5
1880.513 feet
x 145.54t;
y 16t 2 122.17t 5
1117.165 feet
Slide 7 - 45Copyright © 2009 Pearson Education, Inc.
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.
c.
x 145.54t;
y 16t 2 122.17t 5
2234.476 feet
x 145.54t;
y 16t 2 122.17t 5
1105.231 feet
b.
d. x 145.54t;
y 16t 2 122.17t 5
1880.513 feet
x 145.54t;
y 16t 2 122.17t 5
1117.165 feet
Slide 7 - 46Copyright © 2009 Pearson Education, Inc.
Find parametric equations for y = 9x + 5.
a.
b.
c.
d. x t
9; yt 5
x t; yt
9 5
x t
9; yt 5
x 9t; yt 5
Slide 7 - 47Copyright © 2009 Pearson Education, Inc.
Find parametric equations for y = 9x + 5.
a.
b.
c.
d. x t
9; yt 5
x t; yt
9 5
x t
9; yt 5
x 9t; yt 5
Slide 7 - 48Copyright © 2009 Pearson Education, Inc.
Find parametric equations for 0 ≤ t ≤ 2 that define the curve.
a.
b.
c.
d.
x 7sin2t 1
, y 8 cos2t 1
x 7sin2t 1
, y8 cos2t 1
x 8sin2t 1
, y 7 cos2t 1
x 8sin2t 1
, y7 cos2t 1
Slide 7 - 49Copyright © 2009 Pearson Education, Inc.
Find parametric equations for 0 ≤ t ≤ 2 that define the curve.
a.
b.
c.
d.
x 7sin2t 1
, y 8 cos2t 1
x 7sin2t 1
, y8 cos2t 1
x 8sin2t 1
, y 7 cos2t 1
x 8sin2t 1
, y7 cos2t 1