Stewart Precalc 6e Section 11.6: Polar Equations of Conics

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Stewart Precalc 6e Section 11.6: Polar Equations of Conics 0/51
1 Concept - Definition of conic and eccentricity · Level 1
All conics can be described geometrically using a fixed point \(F\) called the ______ and a fixed line \(\ell\) called the ______. For a fixed positive number \(e\) the set of all points \(P\) satisfying \(|P F| / d(P, \ell) = e\) is a ______. If \(e < 1\), the conic is an ______; if \(e = 1\), the conic is a ______; and if \(e > 1\), the conic is a ______. The number \(e\) is called the ______ of the conic.
2 Concept - Polar equation forms of a conic · Level 1
The polar equation of a conic with eccentricity \(e\) has one of the following forms: \(r = \) ______ or \(r = \) ______
3 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\dfrac{2}{3}\), directrix \(x = 3\).
4 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity \(\dfrac{4}{3}\), directrix \(x = -3\).
5 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, directrix \(y = 2\).
6 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(\dfrac{1}{2}\), directrix \(y = -4\).
7 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Hyperbola, eccentricity 4, directrix \(r = 5 \sec \theta\).
8 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.6\), directrix \(r = 2 \csc \theta\).
9 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, vertex at \(\left(5, \dfrac{\pi}{2}\right)\).
10 Skill - Write polar equation of a conic · Level 2
Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Ellipse, eccentricity \(0.4\), vertex at \((2, 0)\).
11 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{6}{1 + \cos \theta}\)
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12 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{2}{2 - \cos \theta}\)
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13 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{3}{1 - 2 \sin \theta}\)
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14 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{5}{3 - 3 \sin \theta}\)
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15 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{12}{3 + 2 \sin \theta}\)
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16 Skill - Identify and graph polar conic · Level 2
Match the polar equation with its graph, identify the conic, and find the eccentricity. \(r = \dfrac{12}{2 + 3 \cos \theta}\)
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17 Skill - Analyze parabola from polar equation · Level 3
\(r = \dfrac{4}{1 - \sin \theta}\)
(a) Show that the conic is a parabola and sketch its graph.
(b) Find the vertex and directrix and indicate them on the graph.

Enter your answer directly below each part above.

18 Skill - Analyze parabola from polar equation · Level 3
\(r = \dfrac{3}{2 + 2 \sin \theta}\)
(a) Show that the conic is a parabola and sketch its graph.
(b) Find the vertex and directrix and indicate them on the graph.

Enter your answer directly below each part above.

19 Skill - Analyze parabola from polar equation · Level 3
\(r = \dfrac{5}{3 + 3 \cos \theta}\)
(a) Show that the conic is a parabola and sketch its graph.
(b) Find the vertex and directrix and indicate them on the graph.

Enter your answer directly below each part above.

20 Skill - Analyze parabola from polar equation · Level 3
\(r = \dfrac{2}{5 - 5 \cos \theta}\)
(a) Show that the conic is a parabola and sketch its graph.
(b) Find the vertex and directrix and indicate them on the graph.

Enter your answer directly below each part above.

21 Skill - Analyze ellipse from polar equation · Level 4
\(r = \dfrac{4}{2 - \cos \theta}\)
(a) Show that the conic is an ellipse, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the ellipse and the lengths of the major and minor axes.

Enter your answer directly below each part above.

22 Skill - Analyze ellipse from polar equation · Level 4
\(r = \dfrac{6}{3 - 2 \sin \theta}\)
(a) Show that the conic is an ellipse, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the ellipse and the lengths of the major and minor axes.

Enter your answer directly below each part above.

23 Skill - Analyze ellipse from polar equation · Level 4
\(r = \dfrac{12}{4 + 3 \sin \theta}\)
(a) Show that the conic is an ellipse, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the ellipse and the lengths of the major and minor axes.

Enter your answer directly below each part above.

24 Skill - Analyze ellipse from polar equation · Level 4
\(r = \dfrac{18}{4 + 3 \cos \theta}\)
(a) Show that the conic is an ellipse, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the ellipse and the lengths of the major and minor axes.

Enter your answer directly below each part above.

25 Skill - Analyze hyperbola from polar equation · Level 4
\(r = \dfrac{8}{1 + 2 \cos \theta}\)
(a) Show that the conic is a hyperbola, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the hyperbola, and sketch the asymptotes.

Enter your answer directly below each part above.

26 Skill - Analyze hyperbola from polar equation · Level 4
\(r = \dfrac{10}{1 - 4 \sin \theta}\)
(a) Show that the conic is a hyperbola, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the hyperbola, and sketch the asymptotes.

Enter your answer directly below each part above.

27 Skill - Analyze hyperbola from polar equation · Level 4
\(r = \dfrac{20}{2 - 3 \sin \theta}\)
(a) Show that the conic is a hyperbola, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the hyperbola, and sketch the asymptotes.

Enter your answer directly below each part above.

28 Skill - Analyze hyperbola from polar equation · Level 4
\(r = \dfrac{6}{2 + 7 \cos \theta}\)
(a) Show that the conic is a hyperbola, and sketch its graph.
(b) Find the vertices and directrix, and indicate them on the graph.
(c) Find the center of the hyperbola, and sketch the asymptotes.

Enter your answer directly below each part above.

29 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{4}{1 + 3 \cos \theta}\)
30 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{8}{3 + 3 \cos \theta}\)
31 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{2}{1 - \cos \theta}\)
32 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{10}{3 - 2 \sin \theta}\)
33 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{6}{2 + \sin \theta}\)
34 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{5}{2 - 3 \sin \theta}\)
35 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{7}{2 - 5 \sin \theta}\)
36 Skill - Identify conic from polar equation · Level 2
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. \(r = \dfrac{8}{3 + \cos \theta}\)
37 Skill - Rotate a polar conic · Level 3
\(r = \dfrac{1}{4 - 3 \cos \theta}\); rotate by \(\theta = \dfrac{\pi}{3}\).
(a) Find the eccentricity and the directrix of the conic.
(b) If this conic is rotated about the origin through the given angle \(\theta\), write the resulting equation.
(c) Draw graphs of the original conic and the rotated conic on the same screen.

Enter your answer directly below each part above.

38 Skill - Rotate a polar conic · Level 3
\(r = \dfrac{2}{5 - 3 \sin \theta}\); rotate by \(\theta = 2 \dfrac{\pi}{3}\).
(a) Find the eccentricity and the directrix of the conic.
(b) If this conic is rotated about the origin through the given angle \(\theta\), write the resulting equation.
(c) Draw graphs of the original conic and the rotated conic on the same screen.

Enter your answer directly below each part above.

39 Skill - Rotate a polar conic · Level 3
\(r = \dfrac{2}{1 + \sin \theta}\); rotate by \(\theta = -\dfrac{\pi}{4}\).
(a) Find the eccentricity and the directrix of the conic.
(b) If this conic is rotated about the origin through the given angle \(\theta\), write the resulting equation.
(c) Draw graphs of the original conic and the rotated conic on the same screen.

Enter your answer directly below each part above.

40 Skill - Rotate a polar conic · Level 3
\(r = \dfrac{9}{2 + 2 \cos \theta}\); rotate by \(\theta = -5 \dfrac{\pi}{6}\).
(a) Find the eccentricity and the directrix of the conic.
(b) If this conic is rotated about the origin through the given angle \(\theta\), write the resulting equation.
(c) Draw graphs of the original conic and the rotated conic on the same screen.

Enter your answer directly below each part above.

41 Skill - Graphing technology investigation · Level 3
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?
42 Skill - Graphing technology investigation · Level 3
(a) Graph the conics \(r = \dfrac{e d}{1 + e \sin \theta}\) 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?

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43 Application - Orbit of the Earth · Level 4
**Orbit of the Earth** The polar equation of an ellipse can be expressed in terms of its eccentricity \(e\) and the length \(a\) of its major axis.
(a) Show that the polar equation of an ellipse with directrix \(x = -d\) can be written in the form \(r = \dfrac{a (1 - e^2)}{1 - e \cos \theta}\) [Hint: Use the relation \(a^2 = e^2 d^2 / (1 - e^2)^2\) given in the proof on page 766.]
(b) 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.

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44 Application - Perihelion and Aphelion · Level 4
**Perihelion and Aphelion** The planets move around the sun in elliptical orbits with the sun at one focus. The positions of a planet that are closest to, and farthest from, the sun are called its **perihelion** and **aphelion**, respectively.
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(a) Use Exercise 43(a) to show that the perihelion distance from a planet to the sun is \(a(1 - e)\) and the aphelion distance is \(a(1 + e)\).
(b) Use the data of Exercise 43(b) to find the distances from the earth to the sun at perihelion and at aphelion.

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45 Application - Orbit of Pluto · Level 3
**Orbit of Pluto** The distance from Pluto to the sun is \(4.43 \times 10^9\) km at perihelion and \(7.37 \times 10^9\) km at aphelion. Use Exercise 44 to find the eccentricity of Pluto's orbit.
46 Discovery - Distance to a Focus · Level 2
**Distance to a Focus** When we found polar equations for the conics, we placed one focus at the pole. It's easy to find the distance from that focus to any point on the conic. Explain how the polar equation gives us this distance.
47 Discovery - Polar Equations of Orbits · Level 2
**Polar Equations of Orbits** When a satellite orbits the earth, its path is an ellipse with one focus at the center of the earth. Why do scientists use polar (rather than rectangular) coordinates to track the position of satellites? [Hint: Your answer to Exercise 46 is relevant here.]
48 Example - Finding a Polar Equation for a Conic · Level 2
Find a polar equation for the parabola that has its focus at the origin and whose directrix is the line \(y = -6\).
49 Example - Identifying and Sketching a Conic · Level 3
A conic is given by the polar equation \(r = \dfrac{10}{3 - 2 \cos \theta}\)
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(a) Show that the conic is an ellipse, and sketch the graph.
(b) Find the center of the ellipse and the lengths of the major and minor axes.

Enter your answer directly below each part above.

50 Example - Identifying and Sketching a Conic (Hyperbola) · Level 3
A conic is given by the polar equation \(r = \dfrac{12}{2 + 4 \sin \theta}\)
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(a) Show that the conic is a hyperbola and sketch the graph.
(b) Find the center of the hyperbola and sketch the asymptotes.

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51 Example - Rotating an Ellipse · Level 3
Suppose the ellipse of Example 2 is rotated through an angle \(\dfrac{\pi}{4}\) about the origin. Find a polar equation for the resulting ellipse, and draw its graph.
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