# 12.5 Conic sections in polar coordinates

 Page 1 / 8
In this section, you will:
• Identify a conic in polar form.
• Graph the polar equations of conics.
• Deﬁne conics in terms of a focus and a directrix.

Most of us are familiar with orbital motion, such as the motion of a planet around the sun or an electron around an atomic nucleus. Within the planetary system, orbits of planets, asteroids, and comets around a larger celestial body are often elliptical. Comets, however, may take on a parabolic or hyperbolic orbit instead. And, in reality, the characteristics of the planets’ orbits may vary over time. Each orbit is tied to the location of the celestial body being orbited and the distance and direction of the planet or other object from that body. As a result, we tend to use polar coordinates to represent these orbits.

In an elliptical orbit, the periapsis is the point at which the two objects are closest, and the apoapsis is the point at which they are farthest apart. Generally, the velocity of the orbiting body tends to increase as it approaches the periapsis and decrease as it approaches the apoapsis. Some objects reach an escape velocity, which results in an infinite orbit. These bodies exhibit either a parabolic or a hyperbolic orbit about a body; the orbiting body breaks free of the celestial body’s gravitational pull and fires off into space. Each of these orbits can be modeled by a conic section in the polar coordinate system.

## Identifying a conic in polar form

Any conic may be determined by three characteristics: a single focus , a fixed line called the directrix    , and the ratio of the distances of each to a point on the graph. Consider the parabola     $\text{\hspace{0.17em}}x=2+{y}^{2}\text{\hspace{0.17em}}$ shown in [link] .

In The Parabola , we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). In this section, we will learn how to define any conic in the polar coordinate system in terms of a fixed point, the focus $\text{\hspace{0.17em}}P\left(r,\theta \right)\text{\hspace{0.17em}}$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.

If $\text{\hspace{0.17em}}F\text{\hspace{0.17em}}$ is a fixed point, the focus, and $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is a fixed line, the directrix, then we can let $\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ be a fixed positive number, called the eccentricity , which we can define as the ratio of the distances from a point on the graph to the focus and the point on the graph to the directrix. Then the set of all points $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}e=\frac{PF}{PD}\text{\hspace{0.17em}}$ is a conic. In other words, we can define a conic as the set of all points $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ with the property that the ratio of the distance from $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}F\text{\hspace{0.17em}}$ to the distance from $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is equal to the constant $\text{\hspace{0.17em}}e.$

For a conic with eccentricity $\text{\hspace{0.17em}}e,$

• if $\text{\hspace{0.17em}}0\le e<1,$ the conic is an ellipse
• if $\text{\hspace{0.17em}}e=1,$ the conic is a parabola
• if $\text{\hspace{0.17em}}e>1,$ the conic is an hyperbola

With this definition, we may now define a conic in terms of the directrix, $\text{\hspace{0.17em}}x=±p,$ the eccentricity $\text{\hspace{0.17em}}e,$ and the angle $\text{\hspace{0.17em}}\theta .$ Thus, each conic may be written as a polar equation , an equation written in terms of $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta .$

## The polar equation for a conic

For a conic with a focus at the origin, if the directrix is $\text{\hspace{0.17em}}x=±p,$ where $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a positive real number, and the eccentricity    is a positive real number $\text{\hspace{0.17em}}e,$ the conic has a polar equation

For a conic with a focus at the origin, if the directrix is $\text{\hspace{0.17em}}y=±p,$ where $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ is a positive real number, and the eccentricity is a positive real number $\text{\hspace{0.17em}}e,$ the conic has a polar equation

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has