SECTION 9.5
9.5
4. Ellipse,
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10. r 2
1 cos 11. r 6
2 sin 12. r 7
2 5 sin 13. r 8
3 3 cos 14. r 10
3 2 sin 15. r 5
2 3 sin 16. r 8
3 cos eccentricity , directrix x 3
eccentricity 12 , directrix y 4
eccentricity 4, directrix r 5 sec eccentricity 0.6, directrix r 2 csc vertex at 5, 2
7. Parabola,
8. Ellipse,
4
1 3 cos 9. r 4
3
directrix y 2
5. Hyperbola,
6. Ellipse,
9–16 ■ (a) Find the eccentricity, (b) identify the conic, (c) give an
equation of the directrix, and (d) sketch the conic.
eccentricity 23 , directrix x 3
3. Parabola,
eccentricity 0.4, vertex at 2, 0
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S Click here for solutions.
1– 8 ■ Write a polar equation of a conic with the focus at the
origin and the given data.
2. Hyperbola,
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CONIC SECTIONS IN POLAR COORDINATES
A Click here for answers.
1. Ellipse,
CONIC SECTIONS IN POLAR COORDINATES
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SECTION 9.5
CONIC SECTIONS IN POLAR COORDINATES
9.5
ANSWERS
E Click here for exercises.
S Click here for solutions.
6
3 + 2 cos θ
2. r =
3. r =
2
1 + sin θ
4. r =
4
2 − sin θ
5. r =
20
1 + 4 cos θ
6. r =
6
5 + 3 sin θ
7. r =
10
1 + sin θ
8. r =
8
5 + 2 cos θ
1. r =
9. (a) 3
12
3 − 4 cos θ
4
3
(b) Hyperbola
(c) x =
(b) Parabola
(c) x = −2
(b) Ellipse
(c) y = 6
(d)
10. (a) 1
13. (a) 1
8
3
(b) Parabola
(c) x =
(b) Ellipse
(c) y = −5
(b) Hyperbola
(c) y = − 35
(b) Ellipse
(c) x = 8
(d)
14. (a) 23
(d)
15. (a) 32
(d)
(d)
11. (a) 12
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(d)
12. (a) 52
(d)
16. (a) 31
(d)
(b) Hyperbola
(c) y = − 75
SECTION 9.5
9.5
CONIC SECTIONS IN POLAR COORDINATES
SOLUTIONS
E Click here for exercises.
1. r =
2
·3
ed
6
3
=
=
1 + e cos θ
3 + 2 cos θ
1 + 23 cos θ
4
·3
12
ed
3
=
2. r =
=
4
1 − e cos θ
3 − 4 cos θ
1 − 3 cos θ
3. r =
ed
1·2
2
=
=
1 + e sin θ
1 + sin θ
1 + sin θ
4. r =
1
·4
4
ed
2
=
=
1 − e sin θ
2 − sin θ
1 − 12 sin θ
5. r = 5 sec θ
r=
(a) e = 1
(b) Parabola
(c) ed = 2 ⇒ d = 2 ⇒ directrix x = −2
⇔ x = r cos θ = 5, so
11. r =
⇔ y = r sin θ = 2, so
·2
6
ed
=
=
r=
3
1 + e sin θ
5 + 3 sin θ
1 + 5 sin θ
7. Focus (0, 0), vertex 5,
π
2
3
1+
(a) e =
3
5
r=
2
1 − cos θ
(d) Vertex (−1, 0) = (1, π)
ed
4·5
20
=
=
1 + e cos θ
1 + 4 cos θ
1 + 4 cos θ
6. r = 2 csc θ
10. r =
1
2
sin θ
1
2
(b) Ellipse
⇒ directrix y = 10 ⇒
(c) ed = 3 ⇒ d = 6 ⇒ directrix y = 6
(d) Vertices 2, π2 and 6, 3π
; center 2, 3π
2
2
ed
10
=
1 + e sin θ
1 + sin θ
8. The directrix is x = 4, so
r=
9. r =
2
·4
ed
8
5
=
=
2
1 + e cos θ
5 + 2 cos θ
1 + 5 cos θ
4
1 + 3 cos θ
12. r =
(a) e = 3
(b) Since e = 3 > 1, the conic is a hyperbola.
(c) ed = 4 ⇒ d =
4
3
⇒ directrix x =
4
3
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(d) The vertices are (1, 0) and (−2, π) = (2, 0); the center
is 32 , 0 ; the asymptotes are parallel to
θ = ± cos−1 − 13 .
7/2
1 − 52 sin θ
(a) e =
5
2
(b) Hyperbola
⇒ d = 75 ⇒ directrix y = − 75
(d) Center 53 , 3π
; vertices − 73 , π2 = 73 , 3π
and
2
2
3π 1, 2
(c) ed =
7
2
■
3
4
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SECTION 9.5
13. r =
CONIC SECTIONS IN POLAR COORDINATES
8/3
1 + cos θ
16. r =
(a) e =
(a) e = 1
8
3
(d) Vertex
14. r =
3
8
3
⇒ directrix x =
8
3
,0
10/3
1 − 23 sin θ
(a) e =
2
3
(b) Ellipse
(c) ed =
10
3
⇒ d = 5 ⇒ directrix y = −5
(d) Vertices 10, π2 and 2, 3π
; center 4, π2
2
15. r =
5/2
1 − 32 sin θ
(a) e =
3
2
(b) Hyperbola
(c) ed =
5
2
⇒
d=
5
3
(d) Vertices −5, π2 = 5,
3π foci (0, 0) and 6, 2
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1
3
(c) ed =
⇒ d=
4
8/3
1
cos θ
3
(b) Ellipse
(b) Parabola
(c) ed =
1+
⇒ directrix y = − 53
3π
2
and 1, 3π
; center 3, 3π
;
2
2
8
3
⇒ d = 8 ⇒ directrix x = 8
(d) Vertices (2, 0) and (4, π); center (−1, 0)