Problem 294 - Calculus Volume 3

Question

Sketch the graph of each conic. $$ r=\frac{32}{3+5 \sin \theta} $$

Step-by-Step Solution

Answer

The graph is a hyperbola, sketched with the pole as the center and directrix near parallel to minor axis.

Step 1: Identify the Type of Conic Section

The given polar equation \( r = \frac{32}{3 + 5 \sin \theta} \) is in the standard form of a conic section, \( r = \frac{ed}{1 + e \sin \theta} \), where the conic has a focus at the pole. Here, comparing coefficients, we have \( e = 5 \), and \( ed = 32 \). Since, \( 1 < e = 5 \), the conic is an ellipse.

Step 2: Determine the Eccentricity

The eccentricity \( e \) determines the shape of the conic. For \( e = 5 \), which is greater than 1, we normally have a hyperbola. However, upon re-evaluation : The given is a mistaken parameter for an hyperbola perception. Let \( e = \frac{5}{3} \). Thus previously calculated conic is elliptical, but adjustment finds a hyperbolic nature when singularly specified.

Step 3: Find the Equation Parameters

Solving for \( d \), we get \( d = \frac{32}{5} \). Now, our equation sits correctly in the hyperbolic format, focusing correction will be \( r = \frac{ed}{e + \sin \theta} \). This properly situates the conic adjustment perception orientation for graphical plot.

Step 4: Sketch the Graph

To sketch the hyperbola, note the following:1. The center of the hyperbola is at the pole.2. The directrix is at the line \( r= \frac{32}{5} \sin \theta = 3 \).3. Plot various points of \( r \) for different values of \( \theta \). With conciseness, some angulation should ideally extend to peer off and present curvature symmetrically.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates. Unlike Cartesian, which uses a grid of horizontal (x) and vertical (y) lines, polar coordinates utilize a central point known as the pole and an angle to describe the position of a point. In polar coordinates:

• The first element, usually represented as \( r \), is the distance from the pole (origin).
• The second element, \( \theta \), is the angle made with the positive x-axis.

This system is particularly useful for describing conic sections that have one focus at the pole. By altering \( r \) with respect to \( \theta \), we can represent curves such as circles, ellipses, parabolas, and hyperbolas. Understanding how these elements interact is crucial for graphing conic sections in polar form.

Graphing Conics

Graphing conic sections involves understanding different forms of equations that describe curves. Conics include circles, ellipses, parabolas, and hyperbolas. In polar coordinates, we describe these using parameters like eccentricity and specific formulas.

• A circle can be expressed as \( r = a \), where every point is equidistant from the pole.
• Ellipses and hyperbolas have formulas such as \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 + e \sin \theta} \), depending on the directrix alignment.
• Sketched graphs require plotting points for different angles \( \theta \) to visualize their forms.

It’s vital to recognize the transformation of mathematical values at different angles to adequately sketch these curves.

Eccentricity

Eccentricity, denoted as \( e \), is a measure used in conic sections that determines the shape of the curve. It quantifies how "stretched" a conic is.

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

For the given equation \( r = \frac{32}{3 + 5 \sin \theta} \), initially, \( e \) was calculated as 5. This induced a re-evaluation since \( e > 1 \) generally indicates a hyperbola, correcting initial ellipse assumptions.

Hyperbola

A hyperbola is a type of conic section that appears as two symmetric, mirror-image curves. When plotted in polar coordinates, it can exhibit an open path where as \( \theta \) approaches certain values, \( r \) increases beyond bound.

• This occurs when the eccentricity \( e > 1 \).
• In polar equations for a hyperbola, the distance \( r \) changes rapidly, becoming much larger as the angles energize the \( \sin \) or \( \cos \) terms in denominators.
• For the given problem, the hyperbola is centered at the pole, and calculations ensure correct orientation.

This section of the conic family contrasts with ellipses due to its diverging arms, allowing it to model phenomena such as hyperbolic geometry and orbital mechanics.