Problem 29 - Precalculus Mathematics for Calculus

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{4}{1+3 \cos \theta}$$

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity 3.

Step 1: Recognize the form of the polar equation

The given equation is \( r = \frac{4}{1 + 3 \cos \theta} \). This is a polar equation of a conic section with a focus at the origin. The general form of a polar conic section's equation is \( r = \frac{ed}{1 + e\cos\theta} \), where \( e \) is the eccentricity and \( d \) is the semi-latus rectum.

Step 2: Identify the eccentricity

In the general equation \( r = \frac{ed}{1 + e\cos\theta} \), compare it with \( r = \frac{4}{1+3 \cos \theta} \). Here, \( ed = 4 \) and \( e = 3 \). Therefore, the eccentricity \( e = 3 \).

Step 3: Identify the type of conic

The eccentricity \( e \) determines the type of conic: \( e = 1 \) is a parabola, \( e < 1 \) is an ellipse, \( e = 0 \) is a circle, and \( e > 1 \) is a hyperbola. Since \( e = 3 > 1 \), the conic is a hyperbola.

Step 4: Sketch the conic

Since it is a hyperbola, the sketch will typically have two branches. However, in polar coordinates, the conic will be centered at the pole with the directrix parallel to the initial line. The vertices correspond where \( \cos \theta eq -\frac{1}{e} \), which typically occur perpendicular to the directrix. Sketch the hyperbola with its major axis along the direction of cosines.

Step 5: Label the vertices

For a hyperbola given as \( r = \frac{ed}{1 + e\cos\theta} \), the vertices are at points where \( \theta = 0 \) or \( \theta = \pi \). Substitute \( \theta = 0 \) and \( \theta = \pi \) to find these points: At \( \theta = 0 \), \( r = \frac{4}{1 + 3} = 1 \). At \( \theta = \pi \), \( r = \frac{4}{1 - 3} = -2 \), but in polar coordinates, negative values indicate a direction. These vertices should be labeled appropriately on the graph.

Key Concepts

Eccentricity

Eccentricity is a fundamental property of conic sections. It helps us understand their shapes and behaviors. Simply put, eccentricity measures how much a conic section deviates from being circular.

Here's how we can categorize different shapes based on eccentricity:

• If the eccentricity \(e = 0\), the shape is a circle – perfectly round.
• If \(0 < e < 1\), it's an ellipse. This looks like a stretched circle.
• If \(e = 1\), it's a parabola. These are the U-shaped curves that get used frequently in physics for things like projectile paths.
• Finally, if \(e > 1\), you're dealing with a hyperbola. This shape has two separate curves, called branches.

Understanding eccentricity is essential when determining the type of conic section you're working with, just as we see in the equation \(r = \frac{4}{1+3 \cos \theta}\) where the eccentricity \(e = 3\). This tells us we are dealing with a hyperbola, as seen by analyzing the parameter \(e\).

Having a grip on eccentricity is the key first step in mastering conics! It guides us in identifying the shape and properties of different conic sections.

Polar Equations

Polar equations are a vital tool for describing conic sections like ellipses, parabolas, and hyperbolas using a coordinate system that's a bit different from the usual x and y setup. In polar coordinates, each point on a plane is determined by a distance from a reference point and an angle from a direction.

The general form of a polar equation for a conic section is given by:\[ r = \frac{ed}{1 + e \cos \theta} \]Here:

• \(r\) is the radius or distance from the origin (or pole),
• \(e\) is the eccentricity, dictating the conic type,
• \(d\) is the semi-latus rectum, a segment of the line perpendicular to the major axis,
• \(\theta\) is the angle from the polar axis.

In the case of our equation, \(r = \frac{4}{1+3 \cos \theta}\), it can be seen that the equation directly shows the relationship between the angle \(\theta\) and the radius \(r\). Each unique \(\theta\) corresponds to a particular \(r\), creating the shape described by the equation.

Using polar equations is incredibly useful for graphing conic sections and understanding their properties in a more visual, angle-based manner.

Hyperbola

A hyperbola is a fascinating conic section characterized by having two separate branches. Unlike circles or ellipses, hyperbolas do not form a closed surface.

The standard rectangular form of a hyperbola's equation is:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]However, in polar coordinates, it takes another shape like our example equation \(r = \frac{4}{1+3 \cos \theta}\). This version uses the origin as a focal point, and the hyperbola extends in opposite directions based on the eccentricity.

Key features of hyperbolas include:

• They consist of two branches that mirror each other.
• The branches are centered on a pair of fixed points called foci.
• Each branch curves away from the other, shaped by the properties outlined at points where the cosine function balances the equation.

In our exercise, when we evaluate it using polar coordinates, we see these features clearly. Remember:

• The vertices of the hyperbola occur at angles \(\theta = 0\) and \(\theta = \pi\). In our case, these correspond to specific points derived from the formula.
• Sketching these helps visualize how the hyperbola looks in the polar system.

Hyperbolas might seem tricky, but they reveal the beauty of symmetrical curves and the power of conic sections in geometry.