Problem 37 - Algebra and Trigonometry

Question

A polar equation of a conic is given. (a) Find the eccentricity and the directrix of the conic. (b) If this conic is rotated about the origin through the given angle , write the resulting equation. (c) Draw graphs of the original conic and the rotated conic on the same screen. $$ r=\frac{1}{4-3 \cos \theta} ; \quad \theta=\frac{\pi}{3} $$

Step-by-Step Solution

Answer

Eccentricity is 3, directrix is \(\frac{1}{3}\); rotated equation: \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \).

Step 1: Identify the Eccentricity

The given polar equation of the conic is \( r = \frac{1}{4 - 3 \cos \theta} \). The general form for a conic in polar coordinates is \( r = \frac{ed}{1 - e \cos \theta} \) for conics with a focus on the positive x-axis. From the equation \( 4 - 3 \cos \theta \), compare it with \( 1 - e \cos \theta \) which suggests that the eccentricity \( e = 3 \).

Step 2: Identify the Directrix

Rewriting the given equation \( r = \frac{1}{4 - 3 \cos \theta} \) into the form \( r = \frac{ed}{1 - e \cos \theta} \), we find \( ed = 1 \). Since \( e = 3 \), the directrix \( d \) is found by solving \( 3d = 1 \), hence \( d = \frac{1}{3} \).

Step 3: Determine Rotated Equation

The conic equation needs to be rotated by \( \theta = \frac{\pi}{3} \). The general formula for rotation of the conic is: if \( r = \frac{ed}{1 - e \cos(\theta - \theta_0)} \) is the original, the rotated form is \( r = \frac{ed}{1 - e \cos(\theta - (\theta_0 - \phi))} \). Substituting \( \phi = \frac{\pi}{3} \), the new equation becomes \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \).

Step 4: Graph the Original and Rotated Conics

Using a graph plotting tool, plot the original equation \( r = \frac{1}{4 - 3 \cos \theta} \) and the rotated one \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \). The original conic will appear as a hyperbola since \( e > 1 \), and the rotated conic will have the same shape but will be oriented differently on the polar plane. Ensure both graphs are displayed on the same screen for comparison.

Key Concepts

Understanding Eccentricity

Eccentricity is a key concept in understanding conic sections, especially when interpreting them in polar coordinates. It is denoted by the symbol \( e \). Imagine eccentricity as a measure of how much a conic section deviates from being circular. In simpler terms, eccentricity defines the shape and type of the conic.

• For a circle, \( e = 0 \).
• For an ellipse, \( 0 < e < 1 \).
• For a parabola, \( e = 1 \).
• For a hyperbola, \( e > 1 \).

Comparing the given polar equation \( r = \frac{1}{4 - 3 \cos \theta} \) with the standard form \( r = \frac{ed}{1 - e \cos \theta} \), we identify the eccentricity as \( e = 3 \). This tells us that our conic section is a hyperbola because its eccentricity is greater than 1.

Learning About the Directrix

Directrix plays an important role in defining the conic section in conjunction with eccentricity. Simply put, the directrix is a fixed line to which the distance from any point on the conic section is compared to the distance to the focus.In polar coordinates, the equation for a conic section takes the form \( r = \frac{ed}{1 - e \cos \theta} \), and the constant \( d \) represents the distance to the directrix. Decoding the given equation, where \( ed = 1 \) and \( e = 3 \), we solve for \( d \) to get \( d = \frac{1}{3} \).
The directrix is often paired with the eccentricity to describe how stretched or compressed a conic section is. Understanding the interplay between the two can deepen your grasp of the geometry involved.

Introduction to Conic Sections

Conic sections are fundamental shapes that arise from slicing a three-dimensional double cone. They include circles, ellipses, parabolas, and hyperbolas. These shapes can be described and analyzed using polar coordinates, providing insight into their properties and relationships.

• When a plane cuts through both nappes of the cone and is not parallel to the axis, a hyperbola is formed.
• If the plane cuts through just one nappe, creating a closed curve, an ellipse results.

In our exercise, based on the equation \( r = \frac{1}{4 - 3 \cos \theta} \), we determined that our conic is a hyperbola, as indicated by its eccentricity, \( e = 3 \). Understanding these shapes helps in different fields like astronomy, where the paths of celestial objects often resemble conic sections.