Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{2}{1-\cos \theta}\)

Step-by-Step Solution

Answer

The conic is a parabola with directrix at \( x = -2 \) and eccentricity \( e = 1 \).

Step 1: Identify the Conic Type

The given equation is in the form \( r = \frac{ed}{1 + e \cos \theta} \). By comparing, we notice this resembles the equation for a conic section in polar coordinates with a single focus. Here, it matches the form \( r = \frac{e}{1 - e \cos \theta} \), indicating it's a conic with focus at the origin.

Step 2: Determine the Conic Parameters

In the given equation \( r = \frac{2}{1 - \cos \theta} \), by comparing it to the standard form \( r = \frac{ed}{1 - e \cos \theta} \), we can deduce that \( ed = 2 \) and \( e = 1 \).

Step 3: Conclude the Conic Type

Since the eccentricity \( e = 1 \), the conic is a parabola. A parabola is defined with an eccentricity value of exactly one.

Step 4: Find the Directrix

For a parabola, the relationship \( ed = 2 \) was found. Using \( e = 1 \), we solve \( 1 \cdot d = 2 \) to get \( d = 2 \). Therefore, the directrix is a line that can be described at \( r = 2 \) or \( x = -2 \) in Cartesian coordinates, which is perpendicular to the polar axis.

Key Concepts

Eccentricity

In the study of conic sections, eccentricity is a fundamental concept that helps define the shape of the conic. Eccentricity, often denoted by the letter \( e \), is a non-negative real number that can classify conic sections into different types:

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

Eccentricity measures how much the conic section deviates from being circular. For example, in a parabola, where \( e = 1 \), the curve is "perfectly balanced" between extending outwards, as found in hyperbolas, and curving around on itself, as seen in circles and ellipses. Understanding eccentricity is key to identifying and working with conic sections.

Parabola

A parabola is a unique type of conic section that is defined by its eccentricity of exactly one. This means that every point on the parabola is equidistant from a fixed point called the focus and a fixed line called the directrix. Parabolas have several interesting properties:

• The axis of symmetry, a vertical line, passes through the focus and is perpendicular to the directrix.
• The vertex, the point where the parabola turns, lies exactly halfway between the focus and the directrix.
• In polar coordinates, a parabola can be represented with the equation \( r = \frac{ed}{1 - e \cos \theta} \), where \( e = 1 \) for parabolas.

Parabolas are not just mathematical constructs. They are found in many real-world settings, such as satellite dishes and suspension bridge cables, where their reflective properties are highly valued.

Polar Coordinates

Polar coordinates offer an alternative to the traditional Cartesian coordinate system, especially useful in scenarios involving curves like conic sections. In the polar system, points are defined by a distance from a reference point (the origin) and an angle from a reference direction:

• \( r \) represents the radial distance from the origin.
• \( \theta \) denotes the angle measured typically from the positive x-axis.

For conics, polar coordinates can succinctly express equations such as the standard polar conic form \( r = \frac{ed}{1 + e \cos \theta} \). This representation is particularly powerful for understanding properties of conics centered at the origin, highlighting aspects like symmetry and focus more straightforwardly than Cartesian coordinates sometimes can.

Directrix

The directrix of a conic section is a crucial line used in the definition and construction of the curve. Each type of conic section has a mathematical relationship with its directrix:

• For a parabola, the directrix is a line parallel to the axis of symmetry and located such that each point on the curve is equidistant to both the directrix and the focus.
• In the standard equation \( r = \frac{ed}{1 - e \cos \theta} \) for conics, the parameter \( d \) represents the distance from the focus to the directrix.

In a parabola, the directrix serves as a critical baseline that helps frame its characteristic U-shape. Understanding the directrix helps in locating other defining features of conics, such as the focus and vertex, and plays a role in constructing these curves.

Focus

The focus is a special point for conic sections, acting as a reference that helps define their structure. Each conic section type incorporates the focus into its definition:

• A parabola has one focus, which is located on the axis of symmetry.
• In ellipses and hyperbolas, there are two foci that help maintain the shape.

The defining property in parabolas is that each point on the curve is equidistant from the focus and the directrix. In polar coordinates, with the focus at the origin, equations of conics are often succinct and reveal symmetrical properties. The focus plays a pivotal role in describing and constructing conics and understanding their optical properties, such as how they reflect light.