Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{1}{1+\sin \theta} $$

Short answer

The conic is a parabola opening upward.

Step-by-step solution

Identify the Type of Conic

The given equation is in the polar form \( r = \frac{1}{1 + \sin \theta} \). This appears similar to the form \( r = \frac{ed}{1 + e\sin\theta} \), which suggests the conic is a limaçon. The closeness of this form to the general polar conic equation \( r = \frac{ed}{1 - e\sin\theta} \) helps us identify the type of conic.

Determine the Eccentricity and Directrix

In the equation \( r = \frac{1}{1 + \sin \theta} \), the coefficient of \( \sin \theta \) is 1, which suggests that the eccentricity \( e = 1 \). The directrix is perpendicular to the polar axis, and in this form, it is a vertical line, but is not directly needed for graph sketching in this context.

Analyze the Graph

Since \( e = 1 \), this is a special case indicating that the conic is a parabola. As the equation is \( r = \frac{1}{1 + \sin \theta} \), it represents a parabola symmetric about the vertical line, opening upward.

Sketch the Graph

To sketch the graph, consider plugging in several values of \( \theta \) such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and check symmetrical properties or patterns. At \( \theta = \frac{\pi}{2} \), \( r \) becomes \( \frac{1}{1+1} = \frac{1}{2} \). As \( \theta \to \frac{3\pi}{2} \), \( r \) approaches infinity, reflecting that the parabola opens upwards.

Key concepts

Understanding Conic Sections

Conic sections are fascinating curves obtained when a plane intersects a cone. They include ellipses, parabolas, and hyperbolas. These curves are defined based on the angle of the intersecting plane:

• Ellipses: Formed when the plane cuts through both nappes of the cone.
• Parabolas: Formed when the plane is parallel to a slant of the cone.
• Hyperbolas: Formed when the plane cuts through each nappe of the cone.

Each type of conic section has its own unique equation and properties. Understanding these properties allows us to graph them effectively. In a polar coordinate system, the equations look a bit different from Cartesian representations, providing insights into their geometric nature based on the position and directrix.

Exploring Eccentricity

Eccentricity is a key feature that helps differentiate between types of conic sections. It is denoted by the letter \( e \). Here's how it defines different conics:

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

In this exercise, the polar equation \( r = \frac{1}{1 + \sin \theta} \) yields an eccentricity of \( e = 1 \), classifying it as a parabola. This insight is integral for sketching accurate graphs and understanding how the shape of the conic changes based on \( e \). Predicting the behavior of the graph becomes straightforward with this information.

The Curious Case of Limaçons

A limaçon is a special type of polar curve, and it often arises in discussions about conic sections when considered with variations in eccentricity. It can take different shapes based on the parameter variations: some look very similar to circles or even have an inward dimple. The general polar form for a limaçon is \( r = rac{ed}{1 + ebsin heta} \), where the limaçon can exhibit a fold, cusp, or loop. In this exercise, given the form \( r = \frac{1}{1 + \sin \theta} \), it initially appears to be a limaçon. However, the conic's characteristics and its derivation in polar forms reveal it further as a parabola in the context of eccentricity, veering from typical limaçon behavior.

The Role of Parabolas in Polar Coordinates

Parabolas in polar coordinates are intriguing because they often show symmetry around an axis. In our exercise, the equation \( r = \frac{1}{1 + \sin \theta} \) indicates a parabola. Parabolas have one distinct focus and a directrix, maintaining the property of equal distance from the focus and directrix. Here,

• Focus: Located at the point from which the curve diverges or converges.
• Directrix: Perpendicular to the polar axis but not essential for direct graph sketching in polar.

Understanding these elements helps in visualizing how the parabola appears in polar coordinates: usually opening in the direction opposite to the directrix. This form reinforces the concept that the graph represents a parabola, due to the eccentricity and equation structure.

Graph Sketching Techniques

Graph sketching in the polar coordinate system involves a unique approach. It starts with identifying the type of conic from the equation. Once determined, use critical points to find key features of the graph such as symmetry and direction. For the equation \( r = \frac{1}{1 + \sin \theta} \):

• Plot points for specific angles like \( \theta = 0, \frac{\pi}{2}, \pi \), and \( \frac{3\pi}{2} \) to understand the shape.
• Notice how \( r \) changes especially when \( \theta = \frac{3\pi}{2} \), revealing extensions or symmetry in the graph.
• Use symmetry cues: If \( r \to \infty \) as \( \theta \to \frac{3\pi}{2} \), it confirms that the parabola opens upwards.

These steps will help you sketch accurate polar graphs efficiently, discerning notable characteristics of the curves.