Problem 283 - Calculus Volume 3

Question

Consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{3}{2-6 \sin \theta} $$

Step-by-Step Solution

Answer

The eccentricity is 3, identifying the conic as a hyperbola.

Step 1: Identify the Formula Type

The given polar equation is \( r = \frac{3}{2 - 6 \sin \theta} \), which resembles the standard polar equation for a conic section \( r = \frac{ed}{1 - e \sin \theta} \), where \( e \) is the eccentricity and \( d \) the semi-latus rectum.

Step 2: Rearrange to Match Standard Form

Rewrite the given equation as \( r = \frac{3}{2 - 6 \sin \theta} = \frac{3/2}{1 - 3 \sin \theta} \) by factoring out a \( 3 \) from the denominator.

Step 3: Identify Eccentricity

Compare our equation \( r = \frac{3/2}{1 - 3 \sin \theta} \) to the standard form \( r = \frac{ed}{1 - e \sin \theta} \). We can see that \( ed = \frac{3}{2} \) and \( e = 3 \). Identify the eccentricity \( e = 3 \).

Step 4: Identify the Conic Type

The eccentricity \( e = 3 \) is greater than 1, which identifies the conic as a hyperbola. In conic sections, \( e > 1 \) correspond to hyperbolas.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. These curves are fundamental geometrical shapes and are of great importance in mathematics. They include:

• Circle: A set of points equidistant from a center point.
• Ellipse: A stretched circle where the sum of distances from any point on the ellipse to two focus points remains constant.
• Parabola: A symmetrical open plane curve where any point is equidistant from a fixed point (focus) and a fixed line (directrix).
• Hyperbola: A set of all points, the difference of whose distances from two fixed foci is constant.

Conic sections can be described using Cartesian coordinates, but also using polar coordinates, which is very useful for certain problems in physics and astronomy. The shape of these conics can be determined by the eccentricity of the curve.

Eccentricity

Eccentricity (\(e\)) is a key parameter that helps to define the shape of a conic section. It determines how much the conic section deviates from being circular.

• For a circle, the eccentricity is 0, as all points are equidistant from the center.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity of exactly 1.
• A hyperbola has an eccentricity greater than 1.

In polar equations of conic sections like \(r = \frac{ed}{1 - e \sin \theta}\), the eccentricity is a crucial factor in determining the type of curve. The larger the eccentricity (specifically for hyperbolas), the more "stretched" or elongated the conical section appears. It helps classify whether the conic is a circle, ellipse, parabola, or hyperbola.

Hyperbola

A hyperbola is a type of conic section characterized by having an eccentricity greater than 1, making it look like an open curve with two disconnected branches. The general form of the polar equation of a hyperbola is represented as \(r = \frac{ed}{1 - e \sin \theta}\) or \(r = \frac{ed}{1 - e \cos \theta}\), depending on its orientation.Hyperbolas have:

• Two foci, unlike an ellipse that has a closed loop.
• Two branches that never meet, creating an infinite shape as opposed to an ellipse.
• Asymptotes, which are straight lines that the hyperbola approaches but never touches.

The concept of eccentricity in polar coordinates allows us to easily identify a hyperbola when the eccentricity \(e\) is greater than 1. This recognition guides the understanding of its properties and significance in various scientific fields, such as astronomy, where they model paths of celestial objects under certain conditions.