Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(x=1 ; \quad e=1\)

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{1}{1 - \cos\theta}\).

Step 1: Identifying the Type of Conic

Since the eccentricity \(e=1\), the conic is a parabola. The polar equation of a conic with focus at the origin is given by \(r = \frac{ed}{1 + e\cos\theta}\) or \(r = \frac{ed}{1 + e\sin\theta}\) depending on the position of the directrix.

Step 2: Analyzing the Directrix

The given directrix is \(x=1\). Since this is a vertical line, the cosine component is used for conversion. In polar coordinates, the directrix \(x=\frac{d}{\cos\theta}\), so \(d=1\).

Step 3: Formulating the Polar Equation

Substitute the values \(e=1\) and \(d=1\) into the polar equation formula. Thus:\[r = \frac{1 \cdot 1}{1 - 1 \cdot \cos\theta} = \frac{1}{1 - \cos\theta}\]

Step 4: Verifying the Equation

This equation \(r = \frac{1}{1 - \cos\theta}\) implies a parabola with a directrix that coincides with the line \(x=1\). The directrix being negative shows symmetry to the line through the pole.

Key Concepts

Conic Sections

Conic sections are an essential part of geometry, representing the intersection of a plane with a cone. They include:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

Each type of conic section has distinct properties. By altering the angle and position of the intersecting plane, different conic shapes are formed. These shapes are prevalent not only in geometric theory but also in real-world applications like planetary orbits and architectural structures.
Using polar equations, we can capture these conics mathematically, where the position of the focus and the eccentricity define the shape.

Eccentricity

Eccentricity (\(e\)) is a measure that determines the shape of a conic section. It quantifies how much a conic section deviates from being circular:

• For circles, \(e=0\)
• For ellipses, \(0
• For parabolas, \(e=1\)
• For hyperbolas, \(e>1\)

A conic's eccentricity gives insight into its openness and shape. In the problem above, the eccentricity being 1 indicates a parabola. Understanding eccentricity is crucial for distinguishing among the different conic sections, especially when given equations or geometric parameters.

Parabola

A parabola is one of the four types of conic sections, and it is defined as the set of all points that are equidistant to a point called the focus and a line called the directrix.
This \(e=1\) conic section forms a symmetric curve with the following characteristics:

• It appears U-shaped in cartesian coordinates.
• Its polar equation can be expressed as \(r = \frac{ed}{1 + e\cos\theta}\)
• It is frequently seen in the path of projectiles and reflective properties.

The unique feature of parabolas makes them highly applicable in physics, engineering, and optics, as they can effectively reflect and focus energy, especially light and sound.

Focus

The focus is an essential point in the construction of conic sections, especially in parabolas. It is a point from which distances are measured along with the directrix to define the conic. In the context of the given exercise, the focus is positioned at the origin of the polar coordinate system.
For parabolas:

• The focus is located at one end of the axis of symmetry.
• This focal point is critical in determining the set of points that qualify as lying on the parabola.

Understanding the notion of a focus can help address tasks that involve finding specific points related to the geometric arrangement, such as in satellite dishes and headlights.

Directrix

The directrix is a straight line used in tandem with the focus to define a conic section. For parabolas, this line exists such that it helps maintain the constant distance rule for all points on the parabola. In the exercise under examination, the directrix is given by the equation \(x=1\), signifying a vertical line.http://web.dev
In polar coordinates:

• The directrix helps position the conic section relative to the origin.
• It influences the general form of the polar equation.

The directrix, as a reference line, simplifies the process of understanding and deriving geometric properties related to conics.