10.6: Conic Sections in Polar Coordinates
- Page ID
- 114102
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 shown in Figure 2.
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
- if the conic is an ellipse
- if the conic is a parabola
- 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
The Polar Equation for a Conic
For a conic with a focus at the origin, if the directrix is where is a positive real number, and the eccentricity is a positive real number the conic has a polar equation
For a conic with a focus at the origin, if the directrix is where is a positive real number, and the eccentricity is a positive real number the conic has a polar equation
How To
Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity.
- Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
- Identify the eccentricity as the coefficient of the trigonometric function in the denominator.
- Compare with 1 to determine the shape of the conic.
- Determine the directrix as if cosine is in the denominator and if sine is in the denominator. Set equal to the numerator in standard form to solve for or
Example 1
Identifying a Conic Given the Polar Form
For each of the following equations, identify the conic with focus at the origin, the directrix, and the eccentricity.
- Answer
For each of the three conics, we will rewrite the equation in standard form. Standard form has a 1 as the constant in the denominator. Therefore, in all three parts, the first step will be to multiply the numerator and denominator by the reciprocal of the constant of the original equation, where is that constant.
-
Since the conic is an ellipse. The eccentricity is and the directrix is
-
Since the conic is a hyperbola. The eccentricity is and the directrix is
-
Because the conic is a parabola. The eccentricity is and the directrix is
-
Try It #1
Identify the conic with focus at the origin, the directrix, and the eccentricity for
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.
Example 2
Graphing a Parabola in Polar Form
Graph
- Answer
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is
Because we will graph a parabola with a focus at the origin. The function has a and there is an addition sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in Table 1 will enable us to see the vertices. See Figure 3.
A B C D undefined
Example 3
Graphing a Hyperbola in Polar Form
Graph
- Answer
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 2, which is
Because so we will graph a hyperbola with a focus at the origin. The function has a term and there is a subtraction sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in Table 2 will enable us to see the vertices. See Figure 5.
A B C D
Example 4
Graphing an Ellipse in Polar Form
Graph
- Answer
First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 5, which is
Because so we will graph an ellipse with a focus at the origin. The function has a and there is a subtraction sign in the denominator, so the directrix is
The directrix is
Plotting a few key points as in Table 3 will enable us to see the vertices. See Figure 6.
A B C D
Try It #2
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.
How To
Given the focus, eccentricity, and directrix of a conic, determine the polar equation.
- Determine whether the directrix is horizontal or vertical. If the directrix is given in terms of we use the general polar form in terms of sine. If the directrix is given in terms of we use the general polar form in terms of cosine.
- Determine the sign in the denominator. If use subtraction. If use addition.
- Write the coefficient of the trigonometric function as the given eccentricity.
- Write the absolute value of in the numerator, and simplify the equation.
Example 5
Finding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix
Find the polar form of the conic given a focus at the origin, and directrix
- Answer
The directrix is so we know the trigonometric function in the denominator is sine.
Because so we know there is a subtraction sign in the denominator. We use the standard form of
and and
Therefore,
Example 6
Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix
Find the polar form of a conic given a focus at the origin, and directrix
- Answer
Because the directrix is we know the function in the denominator is cosine. Because so we know there is an addition sign in the denominator. We use the standard form of
and and
Therefore,
Try It #3
Find the polar form of the conic given a focus at the origin, and directrix
Example 7
Converting a Conic in Polar Form to Rectangular Form
Convert the conic to rectangular form.
- Answer
We will rearrange the formula to use the identities
Try It #4
Convert the conic to rectangular form.
Media
Access these online resources for additional instruction and practice with conics in polar coordinates.
10.5 Section Exercises
Verbal
Explain how eccentricity determines which conic section is given.
If a conic section is written as a polar equation, what must be true of the denominator?
If a conic section is written as a polar equation, and the denominator involves what conclusion can be drawn about the directrix?
If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?
What do we know about the focus/foci of a conic section if it is written as a polar equation?
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.
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
Directrix:
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