Problem 46 - Precalculus

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 parabola is \( r = \frac{8}{1 + \cos(\theta)} \)

Step 1: Identify key information for formula

For a parabola, the eccentricity, e, is always 1. Also since the conic opens to the right from pole, theta' is zero. Our vertex of the parabola is given as (8,0). This point, lies midway between the focus (at the pole) and the directrix, so it is d units away from both. Therefore, we can conclude that d also equals 8 in this case.

Step 2: Substitute values into formula

Substitute e = 1, θ' = 0 and d = 8, into the polar equation for a conic, which is \(r = \frac{ed}{1+e\cos(\theta-\theta')} \). Thus, we get \( r = \frac{8}{1 + \cos(\theta)} \)

Step 3: Simplify equation (if necessary)

In this case, there are no further simplifications that we can do, so the equation \( r = \frac{8}{1 + \cos(\theta)} \) is our final result

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle and position of the intersection, we can form different types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each of these shapes has unique properties and equations that define them.

• Circles and parabolas are special cases where the eccentricity is constant (0 for circles and 1 for parabolas).
• Ellipses and hyperbolas have eccentricities greater than 0 but less than 1, and greater than 1, respectively.

Conic sections are foundational concepts in geometry because they not only appear in various scientific fields but also have practical applications in the real world. For example, the trajectory of objects under the influence of gravity can often form a parabola.

Parabola

A parabola is a specific type of conic section characterized by its single, pointed curve known as the vertex. This curve is symmetrical along its axis and has a distinct 'u' shape. In terms of polar coordinates, a parabola's polar equation can be defined when the focus is located at the pole. The equation, derived from its geometric properties, is \[ r = \frac{ed}{1 + e\cos(\theta)} \] For a parabola, the eccentricity \( e \) equals 1, simplifying the equation to \[ r = \frac{d}{1 + \cos(\theta)} \]

• The vertex of the parabola lies at a point directly opposite the focus.
• The distance from the focus to the vertex is equal to the distance from the vertex to the directrix.

Understanding parabolas is key in fields like physics and engineering where they describe the paths of projectiles and the shape of satellite dishes.

Eccentricity

Eccentricity is a parameter that determines the shape of a conic section. It's a measure of how much a conic section deviates from being circular. The value of eccentricity \( e \) for different conic sections is:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

For parabolas, since the eccentricity is exactly 1, the symmetry and shape simplifies their mathematical representation both in Cartesian and polar coordinates. The concept of eccentricity is not just abstract; it is crucial for calculating orbits of planets and the trajectories of projectiles.

Focus and Directrix

The focus and directrix are fundamental features that verify the definition of any conic section. The focus is a fixed point, while the directrix is a fixed line. For a parabola:

• The distance from any point on the parabola to the focus is equal to its distance to the directrix.
• The vertex lies exactly halfway between the focus and the directrix.

This principle determines the parabolic shape and helps in deriving its equation. In polar coordinates, the focus is typically placed at the origin or pole. The relationship between the focus and directrix in a conic section provides a geometrical construction method used in design and architecture, emphasizing their practical importance in real-life applications.