LESSON
8.4
Answers for the lesson “Graph and Write
Equations of Ellipses”
Skill Practice
5.
y
(0, 7)
(0, 2
1. foci
2. Given vertices (6a, 0) or (0, 6a),
2
co-vertices (0, 6b) or (6b, 0),
then the foci are located at (6c, 0)
or (0, 6c) where c 2 5 a 2 2 b2.
3.
10)
21
(3, 0)
x
y
(0, 2)
}
(4, 0)
(0, 67), (63, 0), (0, 62Ï 10 )
1
21
(2
3, 0)
y
6.
x
(0, 8)
}
(64, 0), (0, 62), (62Ï3 , 0)
4.
y
(0, 5)
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(12, 0)
2
23
(4
x
5, 0)
(10, 0)
2
22
(5
3, 0)
x
}
(612,0), (0, 68), (64Ï5 , 0)
y
7.
(0, 9)
}
(610, 0), (0, 65), (65Ï3 , 0)
(20, 0)
2
24
(
319, 0)
x
}
(620, 0), (0, 69), (6Ï 319 , 0)
Algebra 2
Answer Transparencies for Checking Homework
241
8.
(0, 3
y
21)
y
11.
(0, 15)
(0, 4)
(0, 7)
1
3
21
(6, 0)
22
(3, 0)
x
x
}
(0, 64), (63, 0), (0, 6Ï 7 )
}
(0, 615), (66, 0), (0, 63Ï21 )
y
12.
(0, 5)
y
9.
(0, 6)
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2
(0, 3 3) (3, 0)
21
x
(7, 0)
1
22
(2
x
6, 0)
}
(0, 66), (63, 0), (0, 63Ï3 )
}
y
10.
(67, 0), (0, 65), (62Ï 6 , 0)
(0, 3)
(0, 2 2)
1
22
(1, 0)
x
}
(0, 63), (61, 0), (0, 62Ï2 )
Algebra 2
Answer Transparencies for Checking Homework
242
15. The major axis should be the
y
13.
y-axis, not the x-axis.
(0, 8)
y
2
22
(10, 0)
(6, 0)
x
1
21
(610, 0), (0, 68), (66, 0)
14.
(0, 9)
y
(0, 6 2)
x
}
}
16. a 5 Ï 3 , not 3, and b 5 Ï 2 ,
not 2.
y
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2
24
1
(3, 0)
x
21
}
(0, 69), (63, 0), (0, 66Ï2 )
x
y2
x2
17. } 1 } 5 1
9
25
y2
x2
18. } 1 } 5 1
100
36
y2
x2
19. } 1 } 5 1
81
196
y2
x2
20. } 1 } 5 1
36
16
y2
x2
21. } 1 } 5 1
144
121
Algebra 2
Answer Transparencies for Checking Homework
243
y2
x2
22. } 1 } 5 1
156
400
y2
x2
23. } 1 } 5 1
64
28
2
y2
x2
24. } 1 } 5 1
9
16
y2
x2
25. } 1 } 5 1
81
49
y2
x2
26. } 1 } 5 1
16
25
y2
x2
27. } 1 } 5 1
16
4
y
36.
22
y
37.
2
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y2
x2
28. } 1 } 5 1
121
169
y2
x2
29. } 1 } 5 1
7
16
y2
x2
30. } 1 } 5 1
81
45
y2
x2
31. } 1 } 5 1
175
400
y2
x2
32. } 1 } 5 1
225
289
y2
x2
33. } 1 } 5 1
256
60
y2
x2
34. } 1 } 5 1
1600
1024
x
22
38.
x
y
1
22
x
y
39.
1
21
x
35. B
Algebra 2
Answer Transparencies for Checking Homework
244
y
40.
43.
y
1
4
21
y
41.
24
x
44.
1
24
x
y
x
1
21
y
42.
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x
45. The conic changes from an
2
22
x
ellipse elongated along the y-axis,
to a circle, to an ellipse elongated
along the x-axis.
y
1
21
x
Algebra 2
Answer Transparencies for Checking Homework
245
y2
x2
46. Sample answer: } 1 } 5 1;
49
64
215axa15, 210aya10,
the vertices are (68, 0) and the
co-vertices are (0, 67) which can
be found in the window.
47. Sample answer: By definition,
the value of d1 1 d2 for the point
P(a, 0) is the constant
y2
x2
49. }2 1 }2 5 1,
(92.5)
(77.5)
y2
x2
}2 1 }2 5 1;
(55)
(67.5)
about 11,700aAa22,500
y2
}
x2
50. } 1 } 5 1; 4Ï 33 cm
64
196
Comet Halley
51.
}}
(a 2 (2c))2 1 (0 2 0)2 1
Ï}}
(a 2 c)2 1 (0 2 0)2 5
Ï}
}
2
Ï(a 1 c) 1 Ï(a 2 c)2 5
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a 1 c 1 a 2 c 5 2a. Similarly,
for the point (0, b), this value is
also 2a. Since d15 d2 for this
point, the distance from (0, b) to
1
the point (c, 0) is }2 (2a) 5 a.
0.59
35.3
Sun
y2
x2
Sample answer: }
1}
51
21.1
320.4
52. a. Sample answer: The sum of
the distances from any point
on the ellipse to each airport
remains constant, in this case
600 miles.
A right triangle can be formed
with a right angle at the origin and
the two points (0, b) and (c, 0) as
b. (6225, 0)
the other two vertices. The legs of
c. 600 mi; (6300, 0)
this triangle have lengths b and c,
and the length of the hypotenuse
y2
x2
d. about }2 1 }2 5 1
is the distance from (0, b) to the
198
300
point (c, 0), which was shown to
2
y
x2
}
}
53.
1
51
be a. Therefore, by the
36
4
Pythagorean Theorem,
c 2 1 b 2 5 a 2 or c 2 5 a 2 2 b 2.
Problem Solving
y2
x2
48. }
1 }2 5 1; about 763 km2
2
6
(40.5)
Algebra 2
Answer Transparencies for Checking Homework
246
5. a. about 58.6 m
Mixed Review of
Problem Solving
1. a. 3.92 in.
b. about 20.05 m/sec
c. It increases; as the speed
b. y 2 5 15.69x
c.
increases, the radius of the
circle increases since
y
v2
9.8m
r 5 }.
1
6. They are perpendicular; the
21
x
product of their slopes is 21.
25
7. Sample answer: y 2 5 2}x
2
8. 20 mi;
2
0
2. a. x 2 1 y 2a256
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b. no
c. about 18.8 mi
3. a.
Mercury
29
44
Sun
b. 36.5 million mi, 7.5 million mi
c. Sample answer:
y2
x2
}2 1 }2 5 1
35.7
36.5
y2
x2
4. Sample answer: } 1 } 5 1
36
49
Algebra 2
Answer Transparencies for Checking Homework
247
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9. 2.3 in.;
2
.
3
Algebra 2
Answer Transparencies for Checking Homework
248