Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &(8,0)\end{array}$$

Step-by-Step Solution

Answer

The polar equation of the given parabola with its focus at the pole and vertex (8,0) is \( r = \frac{{-8}}{{1 - cos(\theta)}} \).

Step 1: Identify the Type of Conic

In this exercise, we are given a Parabola. The definition of a parabola is the set of all points that are equidistant from a given point (the focus) and a given line (the directrix). Since the focus is at the pole (0,0), the conic section is centered at the origin.

Step 2: Compute the value of p

The distance p from the vertex to the focus in a parabola is a critical parameter. For the given problem, the vertex is given as (8,0) and the focus is at the pole or origin (0,0), thus p is the distance between the vertex and the focus. The computed value is -8, the negative sign indicates that the direction opens towards the left along the negative x-axis.

Step 3: Write the polar equation for Parabola

The polar form equation for a parabola with the focus at the origin is \( r = \frac{{p}}{{1 - ecos(\theta)}}, \) where \( e \) is the eccentricity, \( \theta \) is the angle, and \( r \) is the radial distance from the origin. For a parabola, its eccentricity \( e = 1 \). Substituting \( p = -8 \) and \( e = 1 \) into the parabolic equation, we get \( r = \frac{{-8}}{{1 - cos(\theta)}} \), which is our desired polar equation.

Key Concepts

Conic Sections

Conic sections are fascinating geometric curves that are formed by the intersection of a plane with a double-napped cone. The basic categories are:

• Parabolas
• Ellipses
• Hyperbolas

These conic sections are defined by specific equations in both Cartesian and polar coordinates. Each conic section has distinctive characteristics based on its shape and the parameters involved, such as eccentricity. In the context of polar coordinates, conic sections can be elegantly defined with their focus located at the origin (or the pole), making the mathematics simpler and more intuitive. The polar equation forms vary slightly depending on whether the conic section is a parabola, ellipse, or hyperbola. Particularly, for parabolas, the polar equation takes the form \( r = \frac{p}{1 - e \cos(\theta)} \), where \( p \) is the distance from the focus to the vertex in the context of our example.

Focus and Directrix

A fundamental concept in understanding conic sections is the focus and directrix. For a parabola, the focus is a fixed point from which the distances to any point on the parabola are measured. The directrix, on the other hand, is a line that helps guide the formation of the parabola. The parabola is defined as the set of all points that are equidistant from the focus and the directrix.

• The focus offers a geometric center of attraction, so to speak, in the parabola.
• The directrix acts as a balancing line, maintaining the parabolic form.

In our case, the focus is at the pole (0,0), simplifying things as it allows the polar form equation to directly involve the parameter \( p \), which is the distance measured from the vertex to the focus. The vertex distance \( p \) directly corresponds to the semi-latus rectum of the conic and crucially defines its size and orientation relative to its focus and directrix.

Eccentricity

Eccentricity is a parameter that subtly modifies the shape of a conic section. It helps in distinguishing between the different types of conic sections:

• Eccentricity \( (e) = 1 \) for parabolas, indicating that the curve forms an open U-shape.
• Eccentricity \( (e) < 1 \) for ellipses, suggesting a closed and oval structure.
• Eccentricity \( (e) > 1 \) for hyperbolas, meaning the shape opens up in two separate, symmetrical curves.

The eccentricity focuses on how much a conic deviates from being circular. In polar equations, knowing the eccentricity simplifies transforming the general conic equations to their specific forms, as seen with our parabola example where \( e = 1 \). This results in the polar equation \( r = \frac{-8}{1 - \cos(\theta)} \), simplifying the form and emphasizing the directionality of the parabola opening.