Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=-4 \cos \theta $$

Step-by-Step Solution

Answer

The curve is a line, represented by \( x = -4 \). The eccentricity concept does not apply.

Step 1: Identify the Polar Equation

The given polar equation is \( r = -4 \cos \theta \). In polar coordinates, such equations can represent conic sections. We start by comparing this to the standard polar form of conics, which includes the equation: \( r = \frac{ed}{1 - e \cos \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \).

Step 2: Compare the Equation to Conic Form

The given equation \( r = -4 \cos \theta \) can be considered a special case of the conic form with \( e = 1 \) (since \( r = e \cdot d \cdot \cos \theta \)). It matches more with this special form due to \( -4 \) being a constant multiplier of \( \cos \theta \). This indicates a conic section degenerates into a line.

Step 3: Determine the Type of Curve

Because the polar equation takes the form \( r = -d \cos \theta \) (where \( d = 4 \)), this signifies a line. In polar coordinates, when \( r = -a \cos \theta \) (or \( r = -a \sin \theta \)), it represents a line through the origin. Hence, \( r = -4 \cos \theta \) is a straight line.

Step 4: Eccentricity and Conic Identification

For a line represented by a polar equation like \( r = -4 \cos \theta \), the concept of eccentricity is moot since it does not form a typical conic section. If treated as a degenerate conic, it would have an undefined eccentricity (considered as infinity in conics' terms).

Step 5: Sketch the Graph

To sketch \( r = -4 \cos \theta \), recall that it represents a vertical line when translated to Cartesian coordinates. The conversion gives us \( x = -4 \). This line is essentially full for all \( \theta \), presenting a vertical line crossing the x-axis at -4.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These intersections can create several types of curves, such as circles, ellipses, parabolas, and hyperbolas. Each has a unique geometric shape and properties:

• Circles: These occur when the plane cuts the cone parallel to its base.
• Ellipses: Result from a plane intersecting at an angle less than the cone's opening.
• Parabolas: Form when the intersecting plane is parallel to the cone's slant edge.
• Hyperbolas: Appear when the plane cuts through both nappes of the cone.

In polar coordinates, conics are described by the equation \( r = \frac{ed}{1 - e \cos \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the directrix. Understanding conics in polar form is crucial because it allows us to represent curves more naturally and directly, especially when dealing with rotations or transformations.

Eccentricity of Conics

Eccentricity \( e \) is a fundamental property of conic sections that characterizes their shapes. It is a non-negative real number that tells us how much a conic deviates from being circular:

• Circle: Eccentricity \( e = 0 \). The circle is the least eccentric conic.
• Ellipse: Eccentricity is between 0 and 1 (\( 0 < e < 1 \)). An ellipse looks like a stretched circle.
• Parabola: Eccentricity \( e = 1 \). Parabolas have a distinct U-shape.
• Hyperbola: Eccentricity \( e > 1 \). Hyperbolas appear as two symmetrical open curves.

For the given equation \( r = -4 \cos \theta \), we're looking at a special case where it doesn't fit the classic definition of a conic but acts as a degenerate form. In such cases, the concept of eccentricity doesn't apply in the standard way, as it's used to describe actual curves rather than degenerate entities like lines or points.

Graph Sketching in Polar Coordinates

Sketching graphs in polar coordinates involves visualizing how the radius \( r \) changes with the angle \( \theta \). Polar equations define the relationship between \( r \) and \( \theta \), allowing us to trace curves in a polar plot:

• Identify the Form: Determine if the equation is conic or another type by examining its structure. For instance, \( r = -4 \cos \theta \) can be taken as a degenerate line.
• Convert to Cartesian: Sometimes, translating it into Cartesian coordinates helps. Here, the equation becomes \( x = -4 \), simplifying the graphing process.
• Plot Points: Choose specific angles, compute \( r \), and plot the points. This method is useful for more detailed curves.

Understanding these steps provides a straightforward roadmap for visualizing polar equations, ensuring that you can accurately represent the curve or line described. For the equation \( r = -4 \cos \theta \), its plot represents a vertical line crossing the x-axis at \(-4\), indicating a direct and simple visualization.