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

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

Step 1: Identify the conic type

The given polar equation is \( r = \frac{1}{4 - 3 \cos \theta} \). A conic in polar form is given by \( r = \frac{ed}{1 + e \cos \theta} \), where \( e \) is the eccentricity and \( d \) is the directrix. First, we rewrite the equation as \( r = \frac{1}{\frac{1}{4} - \frac{3}{4} \cos \theta} \) or \( r = \frac{\frac{1}{4}}{1 - \frac{3}{4} \cos \theta} \). This reveals that \( e = \frac{3}{4} \) and the constant term is \( ed = \frac{1}{4} \). Therefore, directrix \( d = \frac{1/4}{3/4} = \frac{1}{3} \).

Step 2: Determine conic properties

The eccentricity \( e = \frac{3}{4} < 1 \) indicates the conic is an ellipse. The directrix is \( \frac{1}{3} \). Since \( e < 1 \), we confirm the form of the polar equation describes an ellipse.

Step 3: Rotate the conic

We need to find the effect of rotating the conic by angle \( \theta = \frac{\pi}{3} \). The rotation of a polar equation generally transforms \( \cos \theta \) terms into \( \cos(\theta - \alpha) \), where \( \alpha \) is the rotation angle. Hence, the rotated equation is \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \).

Step 4: Sketch the graphs

To graph the conics, we plot the original equation \( r = \frac{1}{4 - 3 \cos \theta} \) and the rotated equation \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \). Graphing both reveals two ellipses: the first aligned with the polar axis, and the second rotated by \( \frac{\pi}{3} \).

Key Concepts

Eccentricity

Eccentricity is a key parameter that helps define and understand conic sections. It acts as a measure of how much a conic section deviates from being circular. Eccentricity, often denoted by the letter \( e \), plays a crucial role in determining the shape of a conic section. There are several categories of conics based on their eccentricity value:

• If \( e = 0 \), the conic is a perfect 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.

In our particular polar equation \( r = \frac{1}{4 - 3 \cos \theta} \), the eccentricity is \( e = \frac{3}{4} \). Since \( \frac{3}{4} < 1 \), we can easily identify the conic as an ellipse. Understanding eccentricity helps in visualizing the geometric nature of the conic section without graphing it directly.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These curves include circles, ellipses, parabolas, and hyperbolas. Each type of conic has distinct properties that relate to its equation. In polar coordinates, the general equation for a conic section is \( r = \frac{ed}{1 + e \cos \theta} \). Here, \( e \) is the eccentricity and \( d \) is the directrix, a fixed line used in the geometric definition of conics.
For our equation \( r = \frac{1}{4 - 3 \cos \theta} \):

• By rearranging, it matches the form \( r = \frac{ed}{1 - e \cos \theta} \), indicating an ellipse.
• The directrix is calculated using \( d = \frac{1}{4} \, / \, \frac{3}{4} = \frac{1}{3} \).

This concept not only aids in classifying the conics but also provides a basis for deriving their geometric properties and equations in different coordinate systems.

Rotation of Conics

Rotating a conic section around the origin involves adjusting the standard polar equation to account for the rotation angle. In practice, when we rotate a conic, we modify the angle in the cosine or sine term by subtracting or adding the rotation angle \( \alpha \). For instance, the cosine term \( \cos \theta \) becomes \( \cos(\theta - \alpha) \) where \( \alpha \) is the angle of rotation.
In our example, we rotate by \( \alpha = \frac{\pi}{3} \). Therefore, the polar equation \( r = \frac{1}{4 - 3 \cos \theta} \) transforms to \( r = \frac{1}{4 - 3 \cos(\theta - \frac{\pi}{3})} \).
Rotation is crucial for aligning conics according to different axes, making it easier to visualize or solve problems where orientation or symmetry matters. Understanding how to apply rotation helps in manipulating and working with conic sections in various mathematical contexts.