Problem 282 - Calculus Volume 3

Question

Consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{5}{-1+2 \sin \theta} $$

Step-by-Step Solution

Answer

Eccentricity is 2; the conic is a hyperbola.

Step 1: Identify Standard Form

The given polar equation is \( r = \frac{5}{-1 + 2 \sin \theta} \). This resembles the standard polar conic form \( r = \frac{ed}{1 + e \sin \theta} \), except the denominator sign. To match the standard form, compare coefficients and rearrange if needed.

Step 2: Compare and Find Eccentricity

The polar equation can be written as \( r = \frac{5}{-(1 - 2 \sin \theta)} \). By rearranging, it becomes \( r = \frac{-5}{1 - 2 \sin \theta} \). Comparing with the standard form \( r = \frac{ed}{1 + e \sin \theta} \), the eccentricity \( e = 2 \).

Step 3: Determine the Type of Conic

Recall that for conic sections, if the eccentricity \( e = 1 \), it represents a parabola; \( e < 1 \), an ellipse; \( e > 1 \), a hyperbola. In this case, since \( e = 2 \), the conic is a hyperbola.

Key Concepts

Eccentricity

Eccentricity is a crucial concept when studying conic sections, particularly in polar equations. It's a measure that helps determine the shape of a conic section. When you look at a conic in polar form, the eccentricity is often denoted by the letter "e." In the context of a polar equation like \( r = \frac{ed}{1 + e \sin \theta} \), the eccentricity gives us a clear numerical value to determine what type of conic we are dealing with.

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), we have an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), it's a hyperbola.

In our given problem, the equation is rearranged to match the standard form, allowing us to extract the eccentricity. In this example, the eccentricity is found to be 2, indicating a hyperbola. This concept helps in categorizing the conic section by simply evaluating one key parameter.

Conic Sections

Conic sections are the different types of curves formed when a plane intersects a double-napped cone. These sections include ellipses, parabolas, hyperbolas, and circles, each with distinct geometric properties. Understanding these different sections is fundamental in geometry and calculus.

• Circle: It is perfectly round, with every point equidistant from the center. It's a special case of an ellipse where eccentricity is zero.
• Ellipse: This is an elongated circle, with eccentricity between 0 and 1. It's more stretched along one axis.
• Parabola: It has a U-shaped curve, with eccentricity equal to 1, characterized by being equidistant from a fixed point (focus) and a line (directrix).
• Hyperbola: This conic section emerges when the plane intersects both nappes of the cone, having eccentricity greater than 1.

In the problem we tackled, we identified the conic section as a hyperbola since the eccentricity was greater than 1. Recognizing these shapes is not only important for identifying conics but also for understanding their distinctive mathematical equations.

Hyperbola

A hyperbola is one of the most fascinating conic sections. It's distinct, as it features two separate curves that open away from each other. Hyperbolas occur naturally in various scientific fields, including astronomy and acoustics, which makes them a practical subject of study in mathematics.

In a hyperbola's polar equation, like \( r = \frac{ed}{1 + e \sin \theta} \), the eccentricity \( e \) must be greater than 1. This signifies that the conic stretches outward beyond an ellipse's bounds.

• Asymptotes: Imaginary lines that the hyperbola approaches but never actually meets. These guide the curve's opening.
• Foci: Two fixed points used in the hyperbola's definition, around which the two branches curve.
• Vertices: The closest points of the two branches, marking the hyperbola's narrowest width.

The given problem had an eccentricity of 2, confirming it's a hyperbola. Understanding and visualizing a hyperbola’s characteristics helps in grasping complex mathematical and real-world phenomena.