Problem 26 - 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. $$ r=\frac{-4}{\cos \theta} $$

Step-by-Step Solution

Answer

The curve is a vertical line \(x = -4\), a degenerate conic.

Step 1: Identify the General Form of Polar Equations

To determine the type of curve given by a polar equation, we need to identify its form. The equation is given as \(r=\frac{-4}{\cos \theta}\), which can be rewritten as \(r=-4\sec \theta\). This is similar to the form \(r=\frac{ed}{1+e\cos\theta}\), where \(e\) is the eccentricity. If we recognize this as a conic with respect to a different standard conic form, \( ed = 4 \).

Step 2: Transform to Cartesian Coordinates

We know \( x = r \cos \theta \) in polar coordinates, so by multiplying both sides of \( r = \frac{-4}{\cos \theta} \) by \( \cos \theta \), we get \( r \cos \theta = -4 \). Therefore, our equation simplifies to \( x = -4 \).

Step 3: Analyze the Cartesian Equation

The Cartesian equation from the polar equation is \( x = -4 \), which represents a vertical line in a plane at a constant distance from the origin.

Step 4: Determine the Type of Curve

Since the Cartesian form \( x = -4 \) is a vertical line, this indicates it is a degenerate conic section (specifically a degenerate parabola). Here, it doesn't describe an ellipse, hyperbola, or a standard parabola.

Step 5: State the Conic and Calculate Eccentricity

Despite the appearance, polar equations of this form typically describe lines, which are represented by a degenerate conic (e.g., zero eccentricity for a line). Here, the eccentricity calculation from the conic form is unnecessary.

Step 6: Sketch the Polar Graph

To sketch the graph of the given polar equation, plot the line \( x = -4 \). This is a straight vertical line that crosses the x-axis at \( x = -4 \), which corresponds to all points where the radius \( r = \frac{-4}{\cos\theta} \) aligns along the line x=-4, consistent with the solution.

Key Concepts

Conic Sections

Conic sections are fundamental curves that arise from the intersection of a plane and a cone. They include the circle, ellipse, parabola, and hyperbola. These shapes are essential in both geometry and algebra. When we talk about conic sections in polar equations, we often see them in forms like

• the ellipse with formula \ r = \frac{ed}{1 + e \cos\theta}
• the hyperbola with formula \ r = \frac{ed}{1 - e \cos\theta}

Each conic is distinct based on its eccentricity \( e \), where different values of \( e \) determine the type of conic:

• \( e = 0 \) for a circle
• \( 0 < e < 1 \) for an ellipse
• \( e = 1 \) for a parabola
• \( e > 1 \) for a hyperbola

Understanding these conics helps us analyze the properties of their polar equations, bringing clarity to the geometric forms they represent.

Cartesian Coordinates

Cartesian coordinates provide a way to describe points and lines on a flat plane using two numbers, often denoted as \( (x, y) \). These coordinates are ideal for representing geometric shapes in algebra because they allow easy plotting and visualization.
In our given problem, the polar equation \( r = \frac{-4}{\cos \theta} \) was transformed into a Cartesian coordinate by recognizing that \( x = r \cos \theta \). This equation was simplified to \( x = -4 \), indicating a straight line.
This transformation helps translate complex polar equations into simpler Cartesian counterparts, enabling a straightforward description of geometrical figures as seen on graph papers or coordinate planes.

Eccentricity

Eccentricity is a parameter that defines the shape of a conic section. It is crucial in distinguishing between different types of conic sections: ellipse, parabola, and hyperbola. Each conic type is characterized by a specific range of eccentricity:

• A circle has an eccentricity \( e = 0 \).
• An ellipse has \( 0 < e < 1 \).
• A parabola has \( e = 1 \).
• A hyperbola features \( e > 1 \).

In the context of the exercise provided, the polar equation led to a degenerate conic. This means the concept of eccentricity is not applicable in the traditional sense here. For lines, considered degenerate conics, eccentricity can be seen as zero, denoting their linear nature compared to other conics.

Degenerate Conic

Degenerate conics are a special case where the intersection of a plane with a cone results in simpler geometric figures. These include:

• Points
• Lines
• Pairs of intersecting lines

In our problem, the transformation of the polar equation \( r = \frac{-4}{\cos\theta} \) to the Cartesian equation \( x = -4 \) hinted at a line, classifying it as a degenerate conic. Typically, such conics do not fit into the standard categories of circle, ellipse, parabola, or hyperbola.
This simplifies their graphical representation. In this instance, the degenerate conic is a straightforward vertical line, showing that even polar equations can sometimes result in unexpected linear constructs.