Problem 55 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(47-56\) $$ r=1 /(1+2 \sin \theta) $$

Step-by-Step Solution

Answer

The graph is a hyperbola with \( e = 2 \), opening upward due to its polar form.

Step 1: Recognize the Conic Section

The given equation is in the form of a polar equation of a conic section: \( r = \frac{ed}{1+e\sin(\theta)} \), where \( e \) is the eccentricity and \( d \) is the semi-latus rectum. From the equation \( r=\frac{1}{1+2\sin(\theta)} \), we identify \( e = 2 \).

Step 2: Determine the Type of Conic

The value \( e = 2 \) is greater than 1, indicating that the conic section is a hyperbola.

Step 3: Rewrite in Standard Form

The given polar equation is already in the standard form of a conic with a focus at the pole: \( r = \frac{1}{1 + 2 \sin(\theta)} \). Here, \( e = 2 \) and \( e > 1 \) confirms that it is a hyperbola. The directrix is perpendicular to the line \( \theta = 0 \), so it is vertical.

Step 4: Plot the Directrix

For this hyperbola, the directrix is a vertical line at distance \( d = -\frac{1}{3} \) from the pole (derived from the distance formula for a hyperbola in polar coordinates), placing it to the left of the pole.

Step 5: Sketch the Asymptotes

Since it is a hyperbola, sketch the asymptotes radiating from the pole at angles where \( 1 + 2 \sin(\theta) = 0 \). These occur when \( \sin(\theta) = -0.5 \), giving angles \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \).

Step 6: Graph the Hyperbola

Plot the hyperbola using the directrix, which helps establish the orientation and symmetry of the graph. The graph opens towards the direction where the sine function is positive. Since \(1 + 2 \sin(\theta)\) is predominant for positive \(\sin(\theta)\), the hyperbola opens upward, confirming reflection symmetry about the directrix.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane in different ways. They include circles, ellipses, parabolas, and hyperbolas. These curves have distinct geometrical properties and are used in various scientific fields.
In a polar equation of a conic section, there are often parameters that define its shape and size. The general form is given by: \[ r = \frac{ed}{1 + e\cos(\theta)}\]It can also use \(\sin(\theta)\), depending on the axis of symmetry.

• The circle and ellipse have an eccentricity \(e < 1\).
• The parabola has an eccentricity \(e = 1\).
• The hyperbola has an eccentricity \(e > 1\).

Conics in polar coordinates help us explore scenarios where angles and distances from a central point (pole) are more significant, such as in planetary orbits and navigation.

Hyperbola

A hyperbola is a type of conic section that forms an open curve with two branches. These branches mirror each other around a central axis, giving the hyperbola its distinct figure.
In a polar equation form, a hyperbola occurs when the eccentricity \(e\) is greater than 1. The equation becomes:\[ r = \frac{ed}{1 + e \sin(\theta)}\]Hyperbolas have unique properties:

• They have two asymptotes, which are lines the curve approaches but never reaches.
• The directrix helps us define the hyperbola's orientation.
• The focal points, situated away from the curve's branches, are key to its definition.

This curve is widely used to describe certain navigational paths and celestial mechanics.

Eccentricity

Eccentricity is a measure of how "stretched" a conic section is. It defines the shape of the conic and its behavior.
Mathematically, eccentricity is represented by the symbol \(e\). For a conic section described by the equation\[ r = \frac{ed}{1 + e \sin(\theta)}\]

• If \(e = 0\), the conic is a circle.
• If \(e = 1\), it is a parabola.
• If \(0 < e < 1\), it forms an ellipse.
• If \(e > 1\), it results in a hyperbola.

The eccentricity reflects how much the conic section departs from being circular. Higher values indicate more elongated shapes.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is defined by a distance from a reference point and an angle from a reference direction. This system contrasts with the rectangular coordinate system, which uses \((x, y)\) values.
In polar coordinates, you specify a point by:

• The radial coordinate \(r\), which is the distance from the pole (origin).
• The angular coordinate \(\theta\), which is the angle from the positive x-axis.

Conic sections are often expressed in polar coordinates to simplify the representation related to circular or elliptical movements. This is especially useful in physics when dealing with orbits and waves. Polar equations like \( r = \frac{ed}{1 + e \sin(\theta)} \) make exploring these scenarios intuitive and straightforward.