Stewart 9e Section 10.6: Conic Sections in Polar Coordinates

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Stewart 9e Section 10.6: Conic Sections in Polar Coordinates 0/32
1 Polar equation of conic from eccentricity and directrix · Level 1
Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\dfrac{1}{2}\), directrix \(x = 4\).
2 Polar equation of parabola from directrix · Level 1
Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix \(x = -3\).
3 Polar equation of hyperbola from eccentricity and directrix · Level 1
Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(1.5\), directrix \(y = 2\).
4 Polar equation of hyperbola from eccentricity and directrix · Level 1
Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(3\), directrix \(x = 3\).
5 Polar equation of ellipse from eccentricity and vertex · Level 2
Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\dfrac{2}{3}\), vertex \((2, \pi)\).
6 Polar equation of ellipse from eccentricity and directrix · Level 2
Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(0.6\), directrix \(r = 4 \csc \theta\).
7 Polar equation of parabola from vertex · Level 2
Write a polar equation of a conic with the focus at the origin and the given data. Parabola, vertex \(\left(3, \dfrac{\pi}{2}\right)\).
8 Polar equation of hyperbola from eccentricity and directrix · Level 2
Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity \(2\), directrix \(r = -2 \sec \theta\).
9 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{4}{5 - 4 \sin \theta} \)
10 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{1}{2 + \sin \theta} \)
11 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{2}{3 + 3 \sin \theta} \)
12 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{5}{2 - 4 \cos \theta} \)
13 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{9}{6 + 2 \cos \theta} \)
14 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{1}{3 - 3 \sin \theta} \)
15 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{3}{4 - 8 \cos \theta} \)
16 Identify conic from polar equation · Level 2
*(a)* Find the eccentricity, *(b)* identify the conic, *(c)* give an equation of the directrix, and *(d)* sketch the conic. \( r = \dfrac{4}{2 + 3 \cos \theta} \)
17 Eccentricity, directrix, and rotation of conic · Level 3
*(a)* Find the eccentricity and directrix of the conic \(r = 1/(1 - 2 \sin \theta)\) and graph the conic and its directrix. *(b)* If this conic is rotated counterclockwise about the origin through an angle \(3 \dfrac{\pi}{4}\), write the resulting equation and graph its curve.
18 Graphing a conic and its rotation · Level 3
Graph the conic \(r = 4/(5 + 6 \cos \theta)\) and its directrix. Also graph the conic obtained by rotating this curve about the origin through an angle \(\dfrac{\pi}{3}\).
19 Effect of eccentricity on conic shape · Level 2
Graph the conics \(r = e/(1 - e \cos \theta)\) with \(e = 0.4, 0.6, 0.8\), and \(1.0\) on a common screen. How does the value of \(e\) affect the shape of the curve?
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20 Effects of parameters d and e on conic shape · Level 2
*(a)* Graph the conics \(r = e d/(1 + e \sin \theta)\) for \(e = 1\) and various values of \(d\). How does the value of \(d\) affect the shape of the conic? *(b)* Graph these conics for \(d = 1\) and various values of \(e\). How does the value of \(e\) affect the shape of the conic?
21 Derivation of polar equation of conic · Level 3
Show that a conic with focus at the origin, eccentricity \(e\), and directrix \(x = -d\) has polar equation \( r = \dfrac{e d}{1 - e \cos \theta} \)
22 Derivation of polar equation of conic · Level 3
Show that a conic with focus at the origin, eccentricity \(e\), and directrix \(y = d\) has polar equation \( r = \dfrac{e d}{1 + e \sin \theta} \)
23 Derivation of polar equation of conic · Level 3
Show that a conic with focus at the origin, eccentricity \(e\), and directrix \(y = -d\) has polar equation \( r = \dfrac{e d}{1 - e \sin \theta} \)
24 Orthogonal intersection of parabolas · Level 3
Show that the parabolas \(r = c/(1 + \cos \theta)\) and \(r = d/(1 - \cos \theta)\) intersect at right angles.
25 Polar equation of Mars's orbit · Level 2
The orbit of Mars around the sun is an ellipse with eccentricity \(0.093\) and semimajor axis \(2.28 \times 10^8\) km. Find a polar equation for the orbit.
26 Polar equation of Jupiter's orbit · Level 2
Jupiter's orbit has eccentricity \(0.048\) and the length of the major axis is \(1.56 \times 10^9\) km. Find a polar equation for the orbit.
27 Halley's comet orbit · Level 3
The orbit of Halley's comet, last seen in 1986 and due to return in 2061, is an ellipse with eccentricity \(0.97\) and one focus at the sun. The length of its major axis is \(36.18\) AU. (An astronomical unit (AU) is the mean distance between the earth and the sun, about \(93\) million miles.) Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?
28 Comet Hale-Bopp orbit · Level 3
Comet Hale-Bopp, discovered in 1995, has an elliptical orbit with eccentricity \(0.9951\). The length of the orbit's major axis is \(356.5\) AU. Find a polar equation for the orbit of this comet. How close to the sun does it come?
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29 Mercury's maximum distance from sun · Level 2
The planet Mercury travels in an elliptical orbit with eccentricity \(0.206\). Its minimum distance from the sun is \(4.6 \times 10^7\) km. Find its maximum distance from the sun.
30 Eccentricity of Pluto's orbit · Level 2
The distance from the dwarf planet Pluto to the sun is \(4.43 \times 10^9\) km at perihelion and \(7.37 \times 10^9\) km at aphelion. Find the eccentricity of Pluto's orbit.
31 Arc length of Mercury's orbit · Level 3
Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around the sun. (If your calculator or computer algebra system evaluates definite integrals, use it. Otherwise, use Simpson's Rule.)
32 Example - Earth's polar orbit equation · Level 2
*(a)* Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about \(0.017\) and the length of the major axis is about \(2.99 \times 10^8\) km. *(b)* Find the distance from the earth to the sun at perihelion and at aphelion.
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