Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-36, 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

A circle with an eccentricity of 0, centered at (-2,0).

Step 1: Recognize the Polar Equation Type

The given polar equation is of the form \(r = a \cos \theta\). This is a type of polar equation that represents a conic section with one of its foci at the origin. The negative sign indicates that the conic is directed along the negative x-axis. This particular form can represent a circle, an ellipse, a parabola, or a hyperbola.

Step 2: Determine if the Conic is a Circle

For a polar equation of the form \(r = a \cos \theta\), the conic is a circle if the absolute value of the coefficient of \(\cos \theta\) or \(\sin \theta\) equals 1. Here, the coefficient is \(-4\), which indicates that this is not a circle. We need to consider other types of conics.

Step 3: Determine the Eccentricity

The general form of a conic section in polar coordinates is \(r = \frac{ep}{1 + e\cos\theta}\) or \(r = \frac{ep}{1 + e\sin\theta}\), where \(e\) is the eccentricity. For the equation given, \(-4 \cos \theta = a(1 + e \cos \theta)\) implies \(a = -4\) and \(e = 0\). Hence, the conic is a circle with an eccentricity of \(0\).

Step 4: Sketch the Graph

Since this is a circle with a radius of \(2\), centered at \((-2, 0)\) on the Cartesian plane (since \(a = -4\) indicates negative x-direction), we place the center of the circle on the negative x-axis, 2 units away from the origin. Draw a circle with this center having a radius of 2.

Key Concepts

Conic Sections

In mathematics, conic sections are the curves obtained by intersecting a right circular cone with a plane. They include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

These shapes are important in various fields such as physics and engineering. Each type is defined by its eccentricity and features distinctive properties.
For example, if the eccentricity is zero, the shape is a circle. As eccentricity increases, the shape could be an ellipse, parabola, or hyperbola. Conic sections thus help in understanding the geometry of these shapes and provide the basis for further analysis of their properties.

Eccentricity

Eccentricity, denoted as \(e\), is a number that describes the shape of a conic section.

• Circle: \(e = 0\): Perfectly round shape.
• Ellipse: \(0 < e < 1\): Stretched circle shape.
• Parabola: \(e = 1\): U-shaped curve.
• Hyperbola: \(e > 1\): Two open curves facing away from each other.

This value indicates how much each conic deviates from being circular. In practical terms, eccentricity is fundamental in determining properties of orbits in celestial mechanics. For the given polar equation \(-4 \cos \theta\), we noticed that the eccentricity is \(0\), indicating a circular shape.

Polar Coordinates

Polar coordinates represent a point in the plane using a distance from a reference point and an angle from a reference direction. It's an alternative to the Cartesian coordinate system.

• Reference Point: Origin.
• Distance from Origin (r): Determines the position along the radial line.
• Angle (\(\theta\)): Measured from the positive x-axis counter-clockwise.

Polar coordinates are particularly useful in cases involving circular or rotational movements. In the given problem, the polar equation \(r = -4 \cos \theta\) describes how the radius \(r\) varies with the angle \(\theta\), which is essential in plotting the curve. Converting polar equations to Cartesian form helps in visualizing them on a standard graph.

Graphing

Graphing polar equations involves plotting points at certain angles and distances from the origin to create shapes in the polar coordinate system. It's crucial for visual understanding of geometric problems.

• Step 1: Identify the form of the polar equation.
• Step 2: Calculate important parameters like eccentricity.
• Step 3: Analyze symmetry and direction.
• Step 4: Plot the points accurately.

In our exercise, we determined this polar equation represents a circle centered at \((-2, 0)\) in the Cartesian plane. Understanding how to move fluidly between polar and Cartesian systems is crucial for graphing with precision.