Problem 37 - Precalculus: Mathematics for Calculus

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 \(\theta\), 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

The eccentricity is 3; the directrix is at a distance of \( \frac{4}{3} \). The rotated conic equation is \( r = \frac{1}{4 - 3\cos(\theta - \frac{\pi}{3})} \).

Step 1: Identify the Polar Equation Components

The given polar equation is \( r = \frac{1}{4 - 3\cos\theta} \). This equation is of the form \( r = \frac{ed}{1 + e\cos\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \) for a conic section. Here, it matches \( r = \frac{ed}{1 - e\cos\theta} \) because the denominator has a minus sign. Let's identify the values: \( e = 3 \) and \( ed = 4 \).

Step 2: Calculate Eccentricity

To find the eccentricity \( e \), we note that it is already given within the equation as \( e = 3 \). Since \( e > 1 \), this conic is a hyperbola.

Step 3: Solve for Directrix Distance

Since we know \( ed = 4 \), we can write the equation to find \( d \):\[ ed = 4 \]Solving for \( d \):\[ 3d = 4 \]\[ d = \frac{4}{3} \]

Step 4: Rotate the Conic

The given rotation angle is \( \theta = \frac{\pi}{3} \). To account for rotation, replace \( \theta \) with \( \theta - \frac{\pi}{3} \) in the original equation:\[ r = \frac{1}{4 - 3\cos\left(\theta - \frac{\pi}{3}\right)} \].

Step 5: Graph the Polar Conic and the Rotated Conic

To graph the original conic, plot \( r = \frac{1}{4 - 3\cos\theta} \). Then, plot the rotated conic using \( r = \frac{1}{4 - 3\cos\left(\theta - \frac{\pi}{3}\right)} \) on the same axes. Observe the differences in orientation between the two graphs.

Key Concepts

Eccentricity

Eccentricity is a measure that determines the shape of a conic section in a polar equation. It is represented by the letter "e" and is pivotal in classifying conics into different categories: circles, ellipses, parabolas, and hyperbolas.

In our example, the polar equation is:

• \( r = \frac{1}{4 - 3 \cos \theta} \)
• This matches the general form for hyperbolas: \( r = \frac{ed}{1 - e\cos \theta} \)
• Identifying the eccentricity, \( e = 3 \), which is already given.

When \( e = 3 \):

• The conic is a hyperbola, as its eccentricity is greater than 1.
• This means that the conic opens in two opposite directions, forming two branches.

Formula for eccentricity can vary based on the type of conic:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

Eccentricity helps categorize and understand the varying nature of conic sections. It's integral to analyzing conic sections in polar coordinates.

Directrix

The directrix of a conic section plays a crucial role in its geometric definition. In the context of polar coordinates, the directrix helps determine the distance characteristics of the conic.

In the given scenario:- The polar equation is \( r = \frac{1}{4 - 3 \cos \theta} \).- Here, the product of eccentricity and directrix distance \((ed)\) is given as 4.- We can solve for the directrix distance, \( d \), using the equation: \[ ed = 4 \]Substituting \( e = 3 \) into the equation: \[ 3d = 4 \] Solving for \( d \), we find:\[ d = \frac{4}{3} \]This distance, \( d \), from the origin, aligns the conic concerning its directrix. In essence, the directrix serves as an important linear reference that influences the shape and position of the conic section with respect to its focus.

Conic Sections

Conic sections are curves formed by intersecting a plane with a double-napped cone. They have different shapes and properties depending on the angle and location of the intersection. In polar coordinates, these sections take on unique characteristics, making them versatile for mathematical representation.

There are four main types of conic sections:

• **Circles**: Symmetrical curves defined as the set of all points equidistant from a center. In polar form with eccentricity = 0.
• **Ellipses**: Oval-shaped curves occurring when the eccentricity is between 0 and 1. Known for having two focal points, with the sum of distances to any point on the ellipse being constant.
• **Parabolas**: Curves that are mirror-symmetrical and where each point is equidistant from a fixed point and a directrix. They have an eccentricity of exactly 1.
• **Hyperbolas**: Formed when the plane intersects both nappes of the cone. These have an eccentricity greater than 1, indicating they have two separate branches.

In our provided polar equation scenario, the conic section is a hyperbola due to its eccentricity being greater than 1 \((e = 3)\). This tells us the curve will open in opposite directions, characteristic of a hyperbola. Understanding these sections helps in visualizing and solving real-world geometry and physics problems involving orbits and trajectories.