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 $P$.

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

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?

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.

$r=\dfrac{1}{1+\cos \theta }$

$r=\dfrac{3}{1-\sin \theta }$

$r=\dfrac{4}{2-3 \sin \theta }$

$r=\dfrac{2}{1+2 \cos \theta }$

$r=\dfrac{3}{4-2 \cos \theta }$

$r=\dfrac{6}{8+2 \cos \theta }$

$r=\dfrac{4}{3+3 \sin \theta }$

$r=\dfrac{1}{6+2 \sin \theta }$

$r=\dfrac{8}{4+3 \sin \theta }$

$r=\dfrac{10}{5+4 \cos \theta }$

$r=\dfrac{9}{3-6 \cos \theta }$

$r=\dfrac{12}{4+8 \sin \theta }$

$r( 3-2 \sin \theta ) =6$

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

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

$r=\dfrac{3\csc \theta }{\csc \theta -1}$

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

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

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

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

$r( 2+\cos \theta ) =4, \theta =\pi$

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

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

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

$e=1;$ directrix is parallel to the polar axis $1$ unit above the pole.

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

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

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

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.

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.

The Effect of Eccentricity

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 $$r=\dfrac{ep}{1+e\cos \theta }$$

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 $r=\dfrac{ep}{1+e\sin \theta }.$

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 $r=\dfrac{ep}{1-e\sin \theta }.$

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.

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 $e$.