Problem 28 - Precalculus with Calculus Previews

Question

Find a polar equation of the parabola with focus at the origin and the given vertex. $$ (2,0) $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 + \cos \theta} \).

Step 1: Understanding the Parabola in Polar Coordinates

A parabola in polar coordinates is generally expressed as \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the directrix. Here, since the focus is at the origin, \( e = 1 \).

Step 2: Defining the Directrix and Vertex Relationship

For a parabola, when the focus is at the origin, the vertex form in polar coordinates is \( r = \frac{p}{1 + \cos \theta} \) with vertex at \((p,0)\). Since our vertex is at \( (2,0) \), here \( p = 2 \).

Step 3: Write the Polar Equation with Known Parameters

Using the derived form \( r = \frac{p}{1 + \cos \theta} \) in step 2 with \( p = 2 \), the polar equation of the parabola is: \( r = \frac{2}{1 + \cos \theta} \).

Key Concepts

Parabola

When talking about a parabola, it's helpful to visualize it as a U-shaped curve. A parabola is defined as the set of all points that are equidistant from a point, called the "focus," and a line, called the "directrix." This unique property gives it a symmetric shape.
In the context of polar coordinates, the parabola can be described using equations that differ from the cartesian representation. The parabola's symmetry, however, remains evident. Understanding how to describe a parabola in both cartesian and polar forms is essential since it allows us to solve problems that involve these curves from different perspectives.
This property relies on the relationship between geometric elements like the focus and directrix, and introduces concepts such as eccentricity, which is crucial for describing conic sections, including parabolas.

Eccentricity

The eccentricity of a conic section is a measure that shows how much it deviates from being circular. For parabolas, the eccentricity is always equal to 1. This means that the parabola is precisely halfway between a flatter ellipse and a more open hyperbola, indicating its unique position among conic sections.
A conic section is uniquely determined by its eccentricity.

• A circle's eccentricity is 0.
• An ellipse's eccentricity is between 0 and 1.
• A parabola's eccentricity is exactly 1.
• A hyperbola's eccentricity is greater than 1.

When discussing parabolas in polar coordinates, the fact that their eccentricity is 1 simplifies certain calculations, as seen in the formula: \[ r = \frac{ed}{1 + e \cos \theta} \]Here, the eccentricity \( e = 1 \) simplifies the equation to a familiar form.

Directrix

The directrix is a crucial component of a parabola and conic sections in general. It is a fixed straight line that helps in defining the parabola along with the focus. For each point on a parabola, the distance to the focus equals the perpendicular distance to the directrix.
In polar coordinates, the directrix can define the shape and orientation of the parabola. With the focus at the origin, as described in the exercise, the directrix runs parallel to the line that passes through the vertex and the origin. The vertex-to-directrix distance is a defining feature of the parabola’s shape and simplifies its equation.

Vertex Form

In polar coordinates, the vertex form of a parabola can take the equation of:\[r = \frac{p}{1 + \cos \theta}\]This formula represents the relationship between the focus and directrix for parabolas. The "vertex" here refers to the point on the parabola that is closest to the origin, which is a specific point of symmetry. The crucial part of using the vertex form is knowing the parameter \( p \), representing the distance from the focus to the vertex.
For example, in the given problem, the vertex is at

• \((2,0)\), meaning \( p = 2 \)

This value directly substitutes into the vertex form equation to provide the polar equation for the parabola, which aligns with the vertex at the specified location.