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Multiple Choice  A(n) [(a) parabola, (b) ellipse, (c) hyperbola] is the set of points \(P\) in the plane for which the distance from a fixed point called the focus to \(P\) equals the distance from a fixed line called the directrix to \(P\).

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In your own words, explain what is meant by the eccentricity \(e\) of a conic.

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Identify the graphs of each of these polar equations: \(r=\dfrac{2}{1+\sin \theta }\) and \(r=\dfrac{2}{1+\cos \theta }.\) How are they the same? How are they different?

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True or False  The polar equation of a conic with focus at the pole and directrix perpendicular to the polar axis at a distance \(p\) to the left of the pole is \(r=\dfrac{ep}{1-p\cos \theta },\) where \(e\) is the eccentricity of the conic.

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\(r=\dfrac{1}{1+\cos \theta }\)

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\(r=\dfrac{3}{1-\sin \theta }\)

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\(r=\dfrac{4}{2-3 \sin \theta }\)

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\(r=\dfrac{2}{1+2 \cos \theta }\)

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\(r=\dfrac{3}{4-2 \cos \theta }\)

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\(r=\dfrac{6}{8+2 \cos \theta }\)

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\(r=\dfrac{4}{3+3 \sin \theta }\)

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\(r=\dfrac{1}{6+2 \sin \theta }\)

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\(r=\dfrac{8}{4+3 \sin \theta }\)

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\(r=\dfrac{10}{5+4 \cos \theta }\)

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\(r=\dfrac{9}{3-6 \cos \theta }\)

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\(r=\dfrac{12}{4+8 \sin \theta }\)

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\(r( 3-2 \sin \theta ) =6\)

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\(r( 2-\cos \theta ) =2\)

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\(r=\dfrac{6 \sec \theta }{2\sec \theta -1}\)

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\(r=\dfrac{3\csc \theta }{\csc \theta -1}\)

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\(r=\dfrac{9}{4-\cos \theta }, \theta =0\)

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\(r=\dfrac{3}{1-\sin \theta }, \theta =0\)

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\(r=\dfrac{8}{4+\sin \theta }, \theta =\dfrac{\pi }{2}\)

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\(r=\dfrac{10}{5+4 \sin \theta }, \theta =\pi\)

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\(r( 2+\cos \theta ) =4, \theta =\pi\)

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\(r( 3-2 \sin \theta ) =6, \theta =\dfrac{\pi }{2}\)

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\(e=\dfrac{4}{5};\) directrix is perpendicular to the polar axis \(3\) units to the left of the pole.

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\(e=\dfrac{2}{3};\) directrix is parallel to the polar axis \(3\) units above the pole.

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\(e=1;\) directrix is parallel to the polar axis \(1\) unit above the pole.

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\(e=1;\) directrix is parallel to the polar axis \(2\) units below the pole.

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\(e=6;\) directrix is parallel to the polar axis \(2\) units below the pole.

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\(e=5;\) directrix is perpendicular to the polar axis \(3\) units to the right of the pole.

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Halley’s Comet As with most comets, Halley’s comet has a highly elliptical orbit about the Sun, given by the polar equation \[ r=\dfrac{1.155}{1-0.967 \cos \theta } \]

where the Sun is at the pole, the semimajor axis is along the polar axis, and \(r\) is measured in AU (astronomical unit). One \(\hbox{AU} =1.5\times 10^{11} {m}\), which is the average distance from Earth to the Sun.

1. What is the eccentricity of the comet’s orbit?
2. Find the distance from Halley’s comet to the Sun at perihelion (shortest distance from the Sun).
3. Find the distance from Halley’s comet to the Sun at aphelion (greatest distance from the Sun).
4. Graph the orbit of Halley’s comet.

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Orbit of Mercury The planet Mercury travels around the Sun in an elliptical orbit given approximately by \[ \begin{equation*} r=\dfrac{\left( 3.442\right) 10^{7}}{1-0.206 \cos \theta } \end{equation*} \]

where \(r\) is measured in miles and the Sun is at the pole.

1. What is the eccentricity of Mercury’s orbit?
2. Find the distance from Mercury to the Sun at perihelion (shortest distance from the Sun).
3. Find the distance from Mercury to the Sun at aphelion (greatest distance from the Sun).
4. Graph the orbit of Mercury.

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The Effect of Eccentricity

1. Graph the conic \(r=\dfrac{2e}{1-e\cos \theta }\) for the following values of \(e\): (i) \(e=0.2,\) (ii) \(e=0.6,\) (iii) \(e=0.9,\) (iv) \(e=1,\) (v) \(e=2\), (vi) \(e=4\).
2. Describe how the shape of the conic changes as \(e>1\) gets larger.
3. Describe how the shape of the conic changes as \(e<1\) gets closer to \(0.\)

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Show that the polar equation for a conic with its focus at the pole and whose directrix is perpendicular to the polar axis at a distance \(p\) units to the right of the pole is given by \[ r=\dfrac{ep}{1+e\cos \theta } \]

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Show that the polar equation for a conic with its focus at the pole and whose directrix is parallel to the polar axis at a distance \(p\) units above the pole is given by \(r=\dfrac{ep}{1+e\sin \theta }.\)

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Show that the polar equation for a conic with its focus at the pole and whose directrix is parallel to the polar axis at a distance \(p\) units below the pole is given by \(r=\dfrac{ep}{1-e\sin \theta }.\)

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Show that the surface area of the solid generated by revolving the first-quadrant arc of the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\;x\geq 0,\;y\geq 0\), about the \(x\)-axis is \[ \begin{equation*} S=\pi b^{2}+\frac{\pi ab}{e}~\sin ^{-1}e \end{equation*} \]

where \(e\) is the eccentricity of the ellipse.

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In this section, one focus of each conic has been at the pole. Write the general equation for a conic in polar coordinates if there is no focus at the pole. That is, suppose the focus \(F\) has polar coordinates \((r_{1}, \theta _{1}),\) and the directrix \(D\) is given by \(r\cos (\theta +\theta _{0})=-d\), where \(d>0\). Let the eccentricity be \(e\).