Problem 281 - Calculus Volume 3

Question

Consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{5}{2+\sin \theta} $$

Step-by-Step Solution

Answer

Eccentricity is 1; the conic is a parabola.

Step 1: Convert Polar Equation to Standard Form

The given polar equation is \( r=\frac{5}{2+\sin \theta} \). The standard form of a conic in polar coordinates is \( r=\frac{ed}{1+e\sin\theta} \) or \( r=\frac{ed}{1+e\cos\theta} \). Here, \( e \) is the eccentricity of the conic, and \( d \) is the semi-latus rectum. By comparing the forms, it looks like the equation is in the form \( r = \frac{ed}{1 + e\sin\theta} \). Identify \( e \) by comparing coefficients in the denominator.

Step 2: Identify the Eccentricity

By comparing \( r=\frac{5}{2+\sin \theta} \) and \( r=\frac{ed}{1+e\sin \theta} \), we see that the denominator in our equation is \( 2 + \sin \theta \). This implies \( e = 1 \) and \( ed = 5 \). Therefore, \( e = 1 \).

Step 3: Determine the Conic Type

Since the eccentricity \( e = 1 \), the conic is a parabola. A parabola is characterized by an eccentricity of exactly 1, and this tells us the conic's shape.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These sections include ellipses, parabolas, and hyperbolas. Each of these shapes has different properties, defined by their eccentricity, which is a measure of how much they deviate from being circular.
Conics can be categorized as follows:

• **Ellipse:** An ellipse is a closed curve that resembles an elongated circle. It has an eccentricity between 0 and 1.
• **Parabola:** A parabola looks like an open curve and has an eccentricity equal to 1.
• **Hyperbola:** A hyperbola consists of two open curves and has an eccentricity greater than 1.

Conic sections are fundamental in geometry and describe various physical phenomena like planetary orbits and the paths of projectiles. Understanding how conic sections are represented, especially in polar coordinates, can aid in solving complex mathematical problems.

Eccentricity

Eccentricity is a key parameter in determining the nature of a conic section. It is denoted by the symbol \( e \) and indicates how much a given conic deviates from a perfect circle.
Here's what eccentricity represents for different conics:

• When \( e = 0 \), the conic is a circle.
• For \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• For \( e > 1 \), the conic is a hyperbola.

Eccentricity is crucial for understanding the properties of conic sections because it determines their shape. In the given exercise, the eccentricity was found to be 1, confirming that the conic is a parabola. This understanding helps us classify conics easily, especially when dealing with equations in forms like polar coordinates.

Polar Coordinates

Polar coordinates offer a unique way to describe the position of a point in a plane using a radius and angle rather than the usual Cartesian coordinates. A point in the polar system is described by \( (r, \theta) \), where \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis.
This system is particularly useful when dealing with conic sections and allows us to express them in a straightforward form. For example, the polar equation \( r = \frac{ed}{1 + e\sin \theta} \) signifies a conic, where \( d \) is the semi-latus rectum and \( e \) is the eccentricity.
Using polar coordinates simplifies problems that feature axial symmetry or circular shapes and is integral to the study of conic sections, such as determining their eccentricity or type based on variations in the polar form. It is a powerful tool in both mathematics and physics, offering insights that are sometimes more complex in Cartesian coordinates.