# Analytic Geometry

## Conic Sections in Polar Coordinates

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 
       shown in .
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 
       at the pole, and a line, the directrix, which is perpendicular to the polar axis.
If 
       is a fixed point, the focus, and 
       is a fixed line, the directrix, then we can let 
       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 
       such that 
       is a conic. In other words, we can define a conic as the set of all points 
       with the property that the ratio of the distance from 
       to 
       to the distance from 
       to 
       is equal to the constant 
      
    
For a conic with eccentricity 
      
    
1. if 
     the conic is an ellipse
2. if 
     the conic is a parabola
3. if 
     the conic is an hyperbola
With this definition, we may now define a conic in terms of the directrix, 
       the eccentricity 
       and the angle 
       Thus, each conic may be written as a polar equation, an equation written in terms of 
       and 
      
    

###  Graphing the Polar Equations of Conics

When graphing in Cartesian coordinates, each conic section has a unique equation. This is not the case when graphing in polar coordinates. We must use the eccentricity of a conic section to determine which type of curve to graph, and then determine its specific characteristics. The first step is to rewrite the conic in standard form as we have done in the previous example. In other words, we need to rewrite the equation so that the denominator begins with 1. This enables us to determine 
       and, therefore, the shape of the curve. The next step is to substitute values for 
       and solve for 
       to plot a few key points. Setting 
       equal to 
       and 
       provides the vertices so we can create a rough sketch of the graph.

###  Deﬁning Conics in Terms of a Focus and a Directrix

So far we have been using polar equations of conics to describe and graph the curve. Now we will work in reverse; we will use information about the origin, eccentricity, and directrix to determine the polar equation.

### Key Concepts

1. Any conic may be determined by a single focus, the corresponding eccentricity,  and the directrix. We can also define a conic in terms of a fixed point, the focus 
         at the pole, and a line, the directrix, which is perpendicular to the polar axis.
2. A conic is the set of all points 
         where eccentricity 
         is a positive real number. Each conic may be written in terms of its polar equation. See .
3. The polar equations of conics can be graphed. See , , and .
4. Conics can be defined in terms of a focus, a directrix, and eccentricity. See  and .
5. We can use the identities 
  and 
  to convert the equation for a conic from polar to rectangular form. See .

### Section Exercises


### Verbal


###  Algebraic

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

###  Extensions

Recall from Rotation of Axes that equations of conics with an 
         term have rotated graphs. For the following exercises, express each equation in polar form with 
         as a function of 
        
      

### Chapter Review Exercises


### The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.
For the following exercises, graph the ellipse, noting center, vertices, and foci.
For the following exercises, use the given information to find the equation for the ellipse.

### The Hyperbola

For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.
For the following exercises,  graph the hyperbola, labeling vertices and foci.
For the following exercises, find the equation of the hyperbola.

### The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.
For the following exercises, graph the parabola, labeling vertex, focus, and directrix.
For the following exercises, write the equation of the parabola using the given information.

### Rotation of Axes

For the following exercises, determine which of the conic sections is represented.
For the following exercises, determine the angle 
         that will eliminate the 
         term, and write the corresponding equation without the 
         term.
For the following exercises, graph the equation relative to the 
         system in which the equation has no 
         term.

### Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

### Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.
For the following exercises, sketch the graph, identifying the center, vertices, and foci.
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle 
       that will eliminate the 
       term.
For the following exercises, rewrite in the 
       system without the 
       term, and graph the rotated graph.
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.