Problem 27 - Essential Calculus

Question

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=-1}\)

Step-by-Step Solution

Answer

The polar equation for the given parabola is \(r = \frac{1}{1 + \cos(\theta)}\).

Step 1: Find distance of directrix from the pole

The directrix solved in the given rectangular form is x = -1. This line lies one unit to the left of the origin. Therefore, its distance from the pole is 1.

Step 2: Substitute values into the general equation

Replace e with 1 (the given eccentricity) and d with 1 (the distance from the pole to the directrix) in the general equation for a parabola. The equation becomes \(r = \frac{1}{1 + \cos(\theta)}\).

Key Concepts

Conic Sections

Conic sections are curves formed by the intersection of a plane with a double-napped cone. These fascinating shapes include the circle, ellipse, parabola, and hyperbola. Each has unique properties defined by their geometric nature.
Conic sections can be identified and classified based on the angle and position of the intersecting plane:

• A circle is formed when the plane cuts the cone parallel to its base.
• An ellipse results when the plane cuts at an angle that is not perpendicular to the cone's axis.
• A parabola occurs when the plane is parallel to the cone's slant height.
• A hyperbola arises when the plane cuts both nappes of the cone.

The study of conic sections is important in many fields, including physics, engineering, and astronomy, due to their natural occurrence and mathematical properties.

Parabola

A parabola is a specific type of conic section characterized by its U-shaped curve. It can be defined as the locus of points equidistant from a fixed point called the focus and a line called the directrix.
Parabolas have several critical features:

• The axis of symmetry, which is the vertical line that divides the parabola into two symmetrical halves.
• The vertex, the point where the parabola changes direction, situated halfway between the focus and the directrix.
• The focus, a point inside the parabola where light reflects off the surface and converges.

The parabola's equation is quite versatile; when represented in polar form, it offers another perspective on its geometry, useful in particular when the focus is at the origin or pole.

Eccentricity

Eccentricity is a fundamental property of conic sections that measures the deviation of a curve from being circular. It is a non-negative real number denoted by the symbol (\(e\)).
Different conic sections have distinct eccentricity values:

• For a circle, the eccentricity is 0.
• An ellipse has eccentricity between 0 and 1.
• A parabola has an eccentricity of exactly 1.
• A hyperbola has an eccentricity greater than 1.

Eccentricity influences the shape, altering how 'stretched' or narrow a conic section appears. For our exercise, the parabola's eccentricity (\(e = 1\)) indicates it is equidistant from its focus to its directrix, yielding its characteristic shape.

Directrix

The directrix is a crucial concept in understanding the properties of conic sections, particularly the parabola. It is a straight line used in conjunction with the focus to formally define conic sections.
In the case of a parabola, every point lies at an equal distance from the focus and the directrix, thus forming its recognizable shape. The directrix is particularly handy when working with polar coordinates. It allows us to derive specific equations that describe the curve.
For our exercise, with a directrix of \(x = -1\), it signifies a vertical line one unit to the left of the pole. Using this directrix, and knowing the parabola's properties, we form the polar equation \( r = \frac{1}{1 + \cos(\theta)} \), offering a succinct and useful mathematical expression.