Problem 32 - Algebra and Trigonometry

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{10}{3-2 \sin \theta} $$

Step-by-Step Solution

Answer

Eccentricity is 2; it's a hyperbola.

Step 1: Identify the General Form

The given equation of the conic is \( r = \frac{10}{3 - 2 \sin \theta} \). This can be rewritten in the form \( r = \frac{ed}{1 - e \sin \theta} \), which is the standard form for conics (specifically, those rotated around horizontal axes).

Step 2: Compare with the Standard Form

By comparing \( r = \frac{10}{3 - 2 \sin \theta} \) with \( r = \frac{ed}{1 - e \sin \theta} \), we identify that the denominator should match. Thus, \( ed = 10 \), \( e = 2 \), and \( d = 5 \).

Step 3: Calculate the Eccentricity

From the comparison, we have \( e = 2 \). This is the eccentricity of the conic.

Step 4: Identify the Type of Conic

The eccentricity \( e = 2 \) is greater than 1, which classifies the conic as a hyperbola.

Step 5: Sketch the Conic

Since the conic is a hyperbola with a vertical directrix because the term contains \( \sin \theta \), sketch two branches symmetric to the directrix. Show the vertices based on the equation \( \theta = 90^\circ \) and \( \theta = 270^\circ \) by substituting these values into the equation to calculate the points where the hyperbola crosses them.

Step 6: Label the Vertices

Calculate the specific points by setting \( r = 0 \), which occurs at specific \( \theta \) if any. However, here we directly use the given formula to identify points for typical angles like \( \theta = 0^\circ, 90^\circ \) directly inserted into the equation; this is challenging for a precise plot without specific graphing software, labeling symbolic vertices instead using calculations.

Key Concepts

Eccentricity

Eccentricity is a key concept in conics that defines the shape of a curve. In the context of conic sections like ellipses, parabolas, and hyperbolas, eccentricity helps determine how "stretched" or "oval" these shapes are when compared to a perfect circle.

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

For the problem at hand, the eccentricity was calculated to be \( e = 2 \), indicating that the figure is a hyperbola. Understanding eccentricity is crucial in identifying the type of conic section you are working with, allowing for accurate sketching and analysis.

Hyperbola

The hyperbola is one of the four classic conic sections, which also include ellipses, parabolas, and circles. Hyperbolas are formed by slicing a double cone with a plane at an angle such that it intersects both halves of the cone. This produces an open curve with two separate branches.

Key features of a hyperbola include:

• Two distinct branches that mirror each other.
• An eccentricity greater than 1, showing it is more spread out than circles or ellipses.
• Asymptotes, which are straight lines that the hyperbola approaches but never touches.
• Vertices, which are the points at the closest point of each branch to the center.

In the exercise provided, the hyperbola was identified due to the eccentricity \( e = 2 \). By identifying such elements, we gain insight into the fundamental structure and behavior of hyperbolas.

Polar Coordinates

Polar coordinates offer a different way to express points in a plane, using a distance from a reference point (the pole) and an angle from a reference direction. This system, in contrast to Cartesian coordinates, is especially useful for dealing with conics when an equation is in the form \( r = f(\theta) \).

In this exercise, the equation \( r = \frac{10}{3 - 2 \sin \theta} \) is presented in polar form.

Key concepts in polar coordinates include:

• \( r \): the radius or distance from the pole.
• \( \theta \): the angle, typically measured in degrees or radians, from the positive x-axis.

This system often simplifies problems involving symmetry and angular motion, making it a powerful tool for sketching and understanding conic sections in polar form.

Sketching Conics

Sketching conics involves visualizing and plotting the geometric shapes defined by specific equations on a coordinate plane. Accurately sketching conics like hyperbolas requires understanding several features extracted from their equations. For example, vertices, axes, and asymptotes play a crucial role in correctly drafting these curves.

To sketch the given hyperbola \( r = \frac{10}{3 - 2 \sin \theta} \), it is helpful to:

• Use key angles like \( 0^\circ \), \( 90^\circ \), \( 180^\circ \), and \( 270^\circ \) to determine salient points.
• Identify symmetry by referring to the directrix, since this equation involves \( \sin \theta \).
• Denote vertices and consider symmetry to draw the branches.

Labeling the vertices accurately on the sketch enhances the representation by highlighting crucial points the curve reaches. These sketches facilitate a deeper understanding of spatial relationships and conic behavior.