Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. $$ r=\frac{4}{1+2 \sin \theta} $$

Step-by-Step Solution

Answer

The curve is a hyperbola with eccentricity 2.

Step 1: Convert Polar Equation to Standard Form

The given equation is \( r = \frac{4}{1 + 2 \sin \theta} \). It resembles the standard form of a conic in polar coordinates, \( r = \frac{ed}{1 + e \sin \theta} \). Here, \( e = 2 \) and \( ed = 4 \), which implies \( d = 2 \). Since \( 1 + e\sin\theta \) indicates the focus is at a point relative to \( \theta = \frac{\pi}{2} \).

Step 2: Identify the Conic Section and Eccentricity

The eccentricity \( e \) is \( 2 \). When \( e > 1 \), the conic is a hyperbola. Thus, this equation represents a hyperbola with an eccentricity of \( 2 \).

Step 3: Sketch the Graph

To sketch the graph, note that in polar equations, \( \sin \theta \) indicates that the vertical axis symmetrical part occurs at \( \theta = \frac{\pi}{2} \). The hyperbola will have one branch toward \(-\theta\) and another around \(\theta\), with a point at the origin when \( \theta = \frac{3\pi}{2} \). Due to limits at which \( r \to \infty \) (as \( \cos \rightarrow -\frac{1}{e} \) becomes undefined), the graph opens downwards and upwards.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a cone with a plane at different angles. These curves include the circle, ellipse, parabola, and hyperbola. Each conic section represents a unique set of geometric properties and serves as fundamental shapes in mathematics. These curves can also be represented using polar coordinates, which are useful when dealing with curves around a common center or focus.

Polar coordinates give us a way to express these curves with equations that relate radius \( r \) and angle \( \theta \). The general form in polar coordinates for a conic is \( r = \frac{ed}{1 + e \sin \theta} \), where \( e \) stands for eccentricity, which determines the shape of the conic. Circles have zero eccentricity, ellipses have eccentricity between zero and one, parabolas have eccentricity exactly one, and hyperbolas have eccentricity greater than one. Understanding the role of eccentricity helps identify which conic section you're dealing with.

Eccentricity

Eccentricity is a central concept in understanding the nature of conic sections. Mathematically, eccentricity \( e \) describes how much a conic section deviates from being a circle. It's a non-negative real number that helps define the shape of the curve:

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

The eccentricity tells us about the stretch of the curve, particularly how far the curve is from being a perfect circle. With hyperbolas, having an eccentricity greater than 1 means that the two branches of the hyperbola are quite widely spaced. In our example, the hyperbola has \( e = 2 \), illustrating a significant deviation from a circular or elliptical shape, which results in the two separate branches of the graph.

Hyperbolas

Hyperbolas are fascinating conic sections defined by their two symmetrical branches. They occur when a plane cuts through both nappes of a cone, resulting in open curves that mirror each other across axes. in polar coordinates, a hyperbola is represented by the equation \( r = \frac{ed}{1 + e \sin \theta} \) if the hyperbola is vertical, or replacing \( \sin \) with \( \cos \) for a horizontal one.

In the given polar equation \( r = \frac{4}{1 + 2 \sin \theta} \), the eccentricity \( e = 2 \) confirms it is a hyperbola. The parameter \( ed = 4 \) indicates the size and spread of the hyperbola. Hyperbolas have the unique property of featuring two branches, which never intersect, opening either up/down or left/right depending on the angle \( \theta \). In the solution, we see that the branches open upwards and downwards, and the focus is aligned with the vertical axis, indicating its symmetrical property relative to \( \theta = \frac{\pi}{2} \). Understanding these properties can help sketch the behavior and asymptotic directions of hyperbolas effectively.