Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{4}{2+2 \cos (\theta-\pi / 3)} $$

Short answer

The curve is a parabola with eccentricity 1.

Step-by-step solution

Identify the Standard Form of the Polar Equation

The given equation is \( r = \frac{4}{2 + 2 \cos(\theta - \pi/3)} \). Notice that it is in the form of a general conic section in polar coordinates: \( r = \frac{ed}{1 - e\cos(\theta - \theta_0)} \), where \( e \) is the eccentricity, and \( d \) is the directrix constant. We need to manipulate the equation to match this form.

Simplify the Given Equation

First, simplify the denominator: \( 2 + 2\cos(\theta - \pi/3) = 2(1 + \cos(\theta - \pi/3)) \). This can be rewritten as:\[r = \frac{4}{2(1 + \cos(\theta - \pi/3))} = \frac{2}{1 + \cos(\theta - \pi/3)}\]

Convert to Standard Form

In the standard form \( r = \frac{ed}{1 - e\cos(\theta - \theta_0)} \), compare and identify the constants. Rewriting \( r = \frac{2}{1 + \cos(\theta - \pi/3)} \) as a conic gives us \( r = \frac{2}{1 - (-1)\cos(\theta - \pi/3)} \), so \( e = 1 \) and \( ed = 2 \).

Determine the Type of Conic and its Eccentricity

Since the eccentricity \( e = 1 \), the conic is a parabola. Parabolas in polar coordinates have an eccentricity of \( e = 1 \). Therefore, the curve described is a parabola, and its eccentricity is 1.

Sketch the Graph

To sketch the graph, note the angle shift of \( \theta - \pi/3 \), which means the parabola is oriented at an angle \( \pi/3 \) from the polar axis. It opens towards the direction of maximum \( \cos(\theta - \pi/3) \), which is along the line \( \theta = \pi/3 \). Draw a parabola opening outwards from the origin along this angle position.

Key concepts

Conic Sections

Conic sections are fundamentally important curves in mathematics, arising from the intersection of a plane with a cone. These curves include circles, ellipses, parabolas, and hyperbolas. Understanding each type of conic section helps in recognizing their distinct properties and equations. In polar coordinates, the polar equation takes a specific form which can describe these sections. You typically see equations like \( r = \frac{ed}{1 - e \cos(\theta - \theta_0)} \), where \( e \) represents the eccentricity and \( d \) the directrix constant.

• A circle has an eccentricity of 0.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity of exactly 1.
• A hyperbola has an eccentricity greater than 1.

These shapes are essential in the study of geometry, physics, and engineering as they help model planets' orbits, signal transmissions, and structural designs.

Eccentricity

Eccentricity is a key characteristic of conic sections that describes how much a conic section deviates from being circular. It's a number that helps us identify the type of conic we are dealing with. The formula \( e = c/a \) defines eccentricity for ellipses and hyperbolas, where \( c \) is the distance from the center to the focus, and \( a \) is the distance from the center to a vertex along the major axis. For the specific polar equation \( r = \frac{4}{2 + 2 \cos(\theta - \pi/3)} \), rewriting it as \( r = \frac{2}{1 + \cos(\theta - \pi/3)} \) reveals that \( e = 1 \), indicating a parabola. This means the shape is perfectly balanced between being an ellipse and a hyperbola, perfectly at the boundary of these two types of conics. Understanding eccentricity is crucial for transitional phenomena in physics and the alignment of orbits in astronomy.

Parabola

The parabola is a unique conic section with the property that every point is equidistant from a fixed point, called the focus, and a fixed line, called the directrix. In polar form, parabolas appear when the eccentricity \( e = 1 \). This particular conic forms a U-shape and is used extensively in various applications, such as satellite dishes and headlight reflectors where focusing light or signals is crucial.For the polar equation given, identify that it changes format to \( r = \frac{2}{1 - (-1)\cos(\theta - \pi/3)} \), confirming the eccentricity \( e \) is 1. This tells us the curve is indeed a parabola. Key features of a parabola—like its axis of symmetry and which direction it opens—can typically be deduced from the equation structure.

Graph Sketching

Graph sketching from polar equations requires understanding directional components covered by angles and rotations. In the exercise, the transformed polar equation \( r = \frac{2}{1 + \cos(\theta - \pi/3)} \) suggests a parabola opening. The term \( \theta - \pi/3 \) indicates a shift or rotation of the parabola, positioning its line of symmetry at \( \theta = \pi/3 \).To accurately sketch:

• Start by drawing the polar axis—a reference line usually horizontal on graph paper.
• Identify the angle \( \pi/3 \) from the polar axis.
• Draw the parabola such that it opens and extends outward along this line.

Graph sketching helps in visualizing how parabolas behave in fields like satellite trajectory calculation and the design of optical devices.