Problem 36 - 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. $$ r=\frac{4}{3 \cos (\theta-\pi / 3)} $$

Step-by-Step Solution

Answer

The curve is a hyperbola with eccentricity \( e = \frac{4}{3} \).

Step 1: Identify the Type of Curve

The given polar equation is \[ r = \frac{4}{3 \cos(\theta - \pi/3)} \]This fits the standard form for polar equations of conics:\[ r = \frac{ed}{1 + e\cos(\theta - \theta_0)} \]Here, the equation suggests a direct relation to this form, indicating a conic centered at an offset angle \(\theta = \pi/3 \) with cosine as the trigonometric function involved. This implies a directrix related to this orientation.

Step 2: Determine Eccentricity and Conic Type

In the general polar equation form for a conic, \[ r = \frac{ed}{1 + e \cos(\theta - \theta_0)} \]where "e" is the eccentricity. By comparing with the given equation, we note:\[ e \times 3 = 4 \]Thus, the eccentricity \( e \) can be calculated as \( e = \frac{4}{3} \). Since the eccentricity \( e > 1 \), this conic is a hyperbola.

Step 3: Understand Conic Graphing

A hyperbola is characterized by two separate curves that mirror each other, originating from points (foci) that are equidistant from a center. The polar equation specifies angular orientation with \( \theta - \pi/3 \), which directs the hyperbola across that axis line.

Step 4: Sketch the Graph

Start by plotting the line associated with \( \theta = \pi/3 \) in a polar grid. Due to the hyperbolic nature and given the equation, the arms of the hyperbola will open away from this line. Since the equation format indicates a hyperbola, expect it to extend into two distinct branches equidistantly reflecting from the focus at the given orientation.

Step 5: Final Verification

Review the calculations and improvements to ensure the eccentricity is pertinent to hyperbolas \( e = \frac{4}{3} \), ensuring accuracy. The outcome aligns with the criteria defining a hyperbolic conic in polar form.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept in the study of conic sections and helps classify the type of conic section that a curve represents. In the context of polar equations, eccentricity, denoted by \( e \), measures how much a conic section deviates from being circular. It is a constant value that determines the shape of the conic:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it's an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

In the given polar equation \( r = \frac{4}{3 \cos(\theta - \pi/3)} \), it's possible to determine the eccentricity by comparing with the standard polar form \( r = \frac{ed}{1 + e \cos(\theta - \theta_0)} \). Here, we find \( e = \frac{4}{3} \), indicating a hyperbola. Understanding eccentricity is crucial because it describes how stretched or compressed the conic section is compared to a circle. This helps in predicting the behavior and orientation of the conic in relation to its focus and directrix.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These sections create different types of curves depending on the angle of the intersection. The main types include:

• Circle: If the plane intersects the cone parallel to its base, we obtain a circle.
• Ellipse: When the intersection is at an angle but does not pass through the base, the result is an ellipse.
• Parabola: A parabola is formed when the plane is parallel to the slant height of the cone.
• Hyperbola: If the plane cuts through both halves of the cone, creating two separate curves, a hyperbola is formed.

For any given polar equation, understanding these conic sections provides a clearer perspective of how the orientation and shape are defined in a coordinate system. In this scenario, the polar equation \( r=\frac{4}{3 \cos(\theta - \pi/3)} \) implies we are dealing with a hyperbola due to the eccentricity \( e > 1 \) and the cosine form indicating a directional orientation.

Hyperbola

A hyperbola is one of the conic sections that occurs when a plane intersects both halves of a double cone. Like other conic sections, hyperbolas have distinct features:

• Foci: Two distinct points on the interior of each curve from which distances are measured to define the hyperbola.
• Asymptotes: Lines that the hyperbola approaches but never meets as it extends infinitely.
• Symmetry: Hyperbolas are symmetric about their center, splitting into two mirror-image branches.

For the polar equation \( r = \frac{4}{3 \cos(\theta - \pi/3)} \), the equation determines that the hyperbola is oriented along the line of \( \theta = \pi/3 \). This means its foci are aligned symmetrically along this axis, and the branches mirror each other across it. The equation's structure informs us about the direction and separation of the branches, all starting from the consideration of its eccentricity \( e = \frac{4}{3} \), verifying the hyperbolic nature of the curve. Understanding these properties of hyperbolas aids in sketching accurate graphs and predicting the path of the curve within its coordinate plane.