Problem 293 - Calculus Volume 3

Question

Sketch the graph of each conic. $$ r=\frac{15}{3-2 \cos \theta} $$

Step-by-Step Solution

Answer

The conic is an ellipse centered at the pole, sketched by plotting points for various \( \theta \) values.

Step 1: Recognize the Type of Conic Section

The given equation is in the form of \( r = \frac{e \cdot d}{1 + e \cdot \cos \theta} \) or \( r = \frac{e \cdot d}{1 - e \cdot \cos \theta} \), which are polar forms of conic sections. Specifically, \( r = \frac{15}{3 - 2 \cos \theta} \) resembles the equation of a conic section with eccentricity \( e \) and directrix \( d \). This indicates it is a conic with a focus at the pole.

Step 2: Identify Eccentricity and Type of Conic

In the equation \( r = \frac{15}{3 - 2 \cos \theta} \), compare it to the standard form \( r = \frac{ed}{1 - e \cos \theta} \). We can identify constants: the numerator suggests \( ed = 15 \) and the term with cosine gives \( e = 2/3 \). Since \( e < 1 \), this conic is an ellipse.

Step 3: Find Directrix (\(d\))

Using the relationship \( ed = 15 \) and \( e = \frac{2}{3} \), solve for \( d \) as follows: \[\frac{2}{3}d = 15 \d = \frac{15 \times 3}{2} \d = 22.5\]

Step 4: Graph the Ellipse

The graph of the ellipse in polar coordinates has its center at the pole (origin). The semi-major axis will lie along the line \( \theta = 0 \) (positive x-axis) because it is related to \( \cos \theta \). Calculate specific points by plugging in convenient values of \( \theta \) to build a sketch. For example, starting with \( \theta = 0, \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2} \).

Step 5: Check Your Sketch

Verify that the calculated points appear symmetrical around the axis \( \theta = \pi \), confirming the ellipse's symmetry around the line where \( \cos \theta \) is involved. Ensure that the graph's eccentricity \( e = \frac{2}{3} \) is less than 1, as expected for an ellipse.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane in various ways. These sections exhibit different shapes, such as circles, ellipses, parabolas, and hyperbolas. The shape depends on the angle at which the plane cuts through the cone.

• **Circles**: These occur when the intersecting plane is perpendicular to the cone's axis.
• **Ellipses**: Formed when the plane cuts through the cone at an angle that results in a closed, oval shape.
• **Parabolas**: These arise when the plane is parallel to the slant of the cone.
• **Hyperbolas**: Resulting from a plane intersecting both nappes (the two parts of the double cone).

By understanding the properties of these conic sections, you can deduce important characteristics such as symmetry, vertices, foci, and axes of symmetry.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. This dimensionless parameter helps classify the type of conic section you are dealing with:

• For a **circle**, the eccentricity is perfectly zero since it is a symmetrical shape.
• **Ellipses** have an eccentricity greater than 0 but less than 1, denoting their oval shape.
• A **parabola** has an eccentricity of exactly 1.
• **Hyperbolas** possess an eccentricity greater than 1, indicating their open curve structure.

In this particular problem, the eccentricity is calculated as \( e = \frac{2}{3} \), confirming that the conic is indeed an ellipse, as \( e < 1 \). This guides us to conclude that the graph will have a closed-loop shape that is not entirely circular.

Ellipse

An ellipse is a set of all points such that the sum of the distances from two fixed points, called foci, is constant. In terms of polar coordinates, the equation can reflect this by using eccentricity values:
When you have an equation like \( r = \frac{ed}{1 - e \cos \theta} \), it shows that the conic is centered around a focus at the pole, with \(e\) being less than 1.
Ellipses have a:

• **Major axis**, which is the longest diameter and passes through both foci.
• **Minor axis**, perpendicular to the major axis at the center.

In this exercise, with \(e = \frac{2}{3}\) and the derived directrix \(d = 22.5\), we define how elongated the ellipse will be compared to a circle. Because the eccentricity fraction isn't too close to zero, the ellipse will have a noticeable flattening but will still maintain a smooth curve.

Graphing Conics

Graphing conics in polar coordinates involves plotting points based on their distance from the origin (pole) and the angle from a reference direction like the positive x-axis. To effectively graph an ellipse like in this task, consider these steps:

• Identify key angles like \( \theta = 0 \), \( \pi/2 \), \( \pi \), and \( 3\pi/2 \). This will help in plotting crucial points on the axes.
• Use the conic equation \( r = \frac{15}{3 - 2 \cos \theta} \) to compute the values of \( r \) at these angles. This determines the ellipse's size and shape.
• Observe symmetry around the major axis, which lies along the line where \( \cos \theta \) influences the equation.

By connecting these points smoothly, the overall shape of the ellipse emerges, ensuring all plotted points maintain the eccentricity condition (\( e = \frac{2}{3} \)) and complete the graph effectively.