In this section we study conic sections in polar coordinates.
Let $F$ be a fixed point (called the focus) and $\ell$ be a fixed line (called the directrix) in a plane. Let $e$ be a fixed positive number (called the eccentricity). The set of all points $P$ in the plane satisfying
$$
\frac{d(P,F)}{d(P,\ell)} = e
$$
is a conic section. The conic is
Recommended Exercise: Read through the proof of this theorem in the book.
Suppose the focus $F$ is at the pole in the polar plane and the directrix $\ell$ is given by $x = d$ for some positive constant $d$. Find equations for each of the three types of conic sections with this focus and this directrix.
Check your answer
$$r = \frac{ed}{1 + e\cos\theta}$$
Video solution
Repeat the previous example for directrix $y = d$.
Check your answer
$$r = \frac{ed}{1 + e\sin\theta}$$
Show that changing the directrix to either $x = -d$ or $y = -d$ results in the following formulas for the conic sections, respectively.
$$
r = \frac{ed}{1 - e\cos\theta}, \ \ \ r = \frac{ed}{1 - e\sin\theta}.
$$
Therefore all conic sections with focus at the pole in the polar plane, and a horizontal or vertical directrix, have formulas of the form
$$
r = \frac{ed}{1 \pm e\cos\theta}, \ \ \text{or} \ \ r = \frac{ed}{1 \pm e\sin\theta}.
$$
Consider the conic section with polar equation
$$
r = \frac{10}{3 - 2\cos\theta}.
$$
Find the eccentricity and directrix, identify the conic, and sketch its graph.
Video solution
Rotate the conic section of the previous example through an angle of $\tfrac{\pi}{3}$. Find a polar function for this rotated conic section.
Answer
$$r = \frac{10}{3 - 2\cos\left(\theta - \tfrac{\pi}{3}\right)}$$
As a beautiful application polar coordinates and conic sections, we end this chapter with a study of Johannes Kepler's Laws of Planetary Motion. Kepler discovered these laws while studying an extensive amount of astronomical data collected by Tycho Brahe.
Kepler's Laws are as follows.
Consider an ellipse
$$
r = \frac{ed}{1 + e\cos\theta}.
$$
Suppose the semi-major-axis of this ellipse has length $a > 0$. Solve for $d$ in terms of $a$ and $e$. Rewrite the equation of the ellipse in terms of $a$ instead of $d$.
Check your answer
$$d = \frac{a(1-e^2)}{e}$$
so that
$$
r = \frac{a(1-e^2)}{1 + e\cos\theta}
$$
Video solution
The positions of a planet that are closest to and farthest from the sun are called its perihelion and aphelion, respectively.
Show that the perihelion and aphelion distances from a planet to the sun are $a(1-e)$ and $a(1+e)$, respectively.
Kepler's Laws apply not only to the orbits of planets around the sun, but to the orbit of any one body around another body. For example, the orbit of the moon around the earth is modeled by Kepler's Laws.
Suppose the semi-major-axis of the moon's orbit around the earth is $385,000$ km. If the eccentricity of the moon's orbit is $0.05$, find the perihelion and aphelion distances of the moon's orbit.
Write a polar equation that models the orbit of the moon around the earth.
Check your answer
The aphelion is $a(1+e) = 385000\cdot 1.05 = 404,250$ km.
The perihelion is $a(1-e) = 385000\cdot 0.95 = 365,750$ km.
The polar equation that models the motion of the moon around the earth is
$$
r = \frac{385000(1-0.0025)}{1 - 0.05\cos\theta} = \frac{384037.5}{1 - 0.05\cos\theta}.
$$
See the orbit
In the image below, the moon's orbit is the solid curve. The dotted curve is the circle centered at the earth with radius $385000$ km. Can you locate the aphelion and perihelion?
Questions? You can ask them below.
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