Analytic Geometry

Conic Sections in Polar Coordinates

Learning Objectives

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 parabolashown in (Figure).

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 focusat the pole, and a line, the directrix, which is perpendicular to the polar axis.

Ifis a fixed point, the focus, andis a fixed line, the directrix, then we can letbe 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 pointssuch thatis a conic. In other words, we can define a conic as the set of all pointswith the property that the ratio of the distance fromtoto the distance fromtois 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 ofand

The Polar Equation for a Conic

For a conic with a focus at the origin, if the directrix is whereis 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.

1. Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
2. Identify the eccentricityas the coefficient of the trigonometric function in the denominator.
3. Comparewith 1 to determine the shape of the conic.
4. Determine the directrix asif cosine is in the denominator andif sine is in the denominator. Setequal to the numerator in standard form to solve foror

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.

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, whereis that constant.

1. Multiply the numerator and denominator by

Because is in the denominator, the directrix isComparing to standard form, note thatTherefore, from the numerator,

Since the conic is an ellipse. The eccentricity isand the directrix is

2. Multiply the numerator and denominator by

Becauseis in the denominator, the directrix isComparing to standard form,Therefore, from the numerator,

Since the conic is a hyperbola. The eccentricity isand the directrix is

3. Multiply the numerator and denominator by

Because sine is in the denominator, the directrix isComparing to standard form,Therefore, from the numerator,

Because the conic is a parabola. The eccentricity isand the directrix is[/hidden-answer]

Try It

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 determineand, therefore, the shape of the curve. The next step is to substitute values forand solve forto plot a few key points. Settingequal to andprovides the vertices so we can create a rough sketch of the graph.

Graphing a Parabola in Polar Form

Graph

First, we rewrite the conic in standard form by multiplying the numerator and denominator by the reciprocal of 3, which is

Becausewe 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 (Figure) will enable us to see the vertices. See (Figure).

A B C D
undefined

Analysis

We can check our result with a graphing utility. See (Figure).

Graphing a Hyperbola in Polar Form

Graph

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 aterm and there is a subtraction sign in the denominator, so the directrix is

The directrix is

Plotting a few key points as in (Figure) will enable us to see the vertices. See (Figure).

A B C D

Graphing an Ellipse in Polar Form

Graph

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 (Figure) will enable us to see the vertices. See (Figure).

A B C D

Analysis

We can check our result using a graphing utility. See (Figure).

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.

1. 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.
3. Write the coefficient of the trigonometric function as the given eccentricity.
4. Write the absolute value of in the numerator, and simplify the equation.

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

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

andand

Therefore,

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

Because the directrix iswe know the function in the denominator is cosine. Becauseso we know there is an addition sign in the denominator. We use the standard form of

andand

Therefore,

Try It

Find the polar form of the conic given a focus at the origin, and directrix

Converting a Conic in Polar Form to Rectangular Form

Convert the conicto rectangular form.

We will rearrange the formula to use the identities

Try It

Convert the conicto rectangular form.

Access these online resources for additional instruction and practice with conics in polar coordinates.

Visit this website for additional practice questions from Learningpod.

Key Concepts

• 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 focusat the pole, and a line, the directrix, which is perpendicular to the polar axis.
• A conic is the set of all points where eccentricityis a positive real number. Each conic may be written in terms of its polar equation. See (Figure).
• The polar equations of conics can be graphed. See (Figure), (Figure), and (Figure).
• Conics can be defined in terms of a focus, a directrix, and eccentricity. See (Figure) and (Figure).
• We can use the identitiesandto convert the equation for a conic from polar to rectangular form. See (Figure).

Section Exercises

Verbal

Explain how eccentricity determines which conic section is given.

If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

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 involveswhat conclusion can be drawn about the directrix?

The directrix will be parallel to the polar axis.

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?

One of the foci will be located at the origin.

Algebraic

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.

Parabola withand directrixunits below the pole.

Hyperbola withand directrixunits above the pole.

Parabola withand directrixunits to the right of the pole.

Ellipse withand directrixunits to the right of the pole.

Hyperbola withand directrixunits above the pole.

Hyperbola withand directrixunits to the right of the pole.

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 anterm have rotated graphs. For the following exercises, express each equation in polar form withas 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.

center:vertices:foci:

For the following exercises, graph the ellipse, noting center, vertices, and foci.

center:vertices:foci:

For the following exercises, use the given information to find the equation for the ellipse.

Center at focus atvertex at

Center atvertex atfocus at

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

Approximately 35.71 feet

The Hyperbola

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

center:vertices:foci:

center:vertices:foci:

For the following exercises, graph the hyperbola, labeling vertices and foci.

For the following exercises, find the equation of the hyperbola.

Center atvertex atfocus at

Foci atandvertex at

The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

vertex:focus:directrix:

vertex:focus: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.

Focus at directrix is

Focus atdirectrix is

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

For the following exercises, determine the anglethat will eliminate theterm, and write the corresponding equation without theterm.

For the following exercises, graph the equation relative to thesystem in which the equation has noterm.

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.

Hyperbola withand directrixunits to the left of the pole.

Ellipse withand directrixunit above the pole.

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.

Directrix isand eccentricity

Directrix isand eccentricity

Practice Test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

center:vertices:foci:

For the following exercises, sketch the graph, identifying the center, vertices, and foci.

center:vertices:foci:

Write the standard form equation of an ellipse with a center atvertex atand focus at

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

center:verticesfoci:asymptotes:

For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

center:vertices:foci:asymptotes:

Write the standard form equation of a hyperbola with foci atandand a vertex at

For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

vertex:focus:directrix:

For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

Write the equation of a parabola with a focus atand directrix

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximatelyfeet

For the following exercises, determine which conic section is represented by the given equation, and then determine the anglethat will eliminate theterm.

For the following exercises, rewrite in thesystem without theterm, and graph the rotated graph.

For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

Hyperbola withand directrixunits to the right of the pole.

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.

Find a polar equation of the conic with focus at the origin, eccentricity ofand directrix:

Glossary

eccentricity
the ratio of the distances from a pointon the graph to the focusand to the directrixrepresented bywhereis a positive real number
polar equation
an equation of a curve in polar coordinatesand