Problem 20 - Calculus Early Transcendentals

Question

(a) Graph the conics \(r=e d /(1+e \sin \theta)\) for \(e=1\) and various values of \(d .\) How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\) . How does the value of \(e\) affect the shape of the conic?

Step-by-Step Solution

Answer

Increasing \(d\) enlarges the parabola. Changing \(e\) modifies the conic: ellipse, parabola, or hyperbola.

Step 1: Understanding the Conic Equation

The given polar equation is \( r = \frac{ed}{1 + e\sin\theta} \). Here, \(e\) is the eccentricity and \(d\) is the semi-latus rectum. For this type of equation, different values of \(e\) and \(d\) will produce different conic sections.

Step 2: Graphing for e=1, Varying d

Set \(e=1\), which indicates a parabola, as \(e\) equals 1. Then vary \(d\): - When \(d=0.5\), plot the graph of \( r = \frac{0.5}{1 + \sin\theta} \).- When \(d=1\), plot \( r = \frac{1}{1 + \sin\theta} \).- When \(d=2\), plot \( r = \frac{2}{1 + \sin\theta} \).Observe that increasing \(d\) enlarges the parabola and shifts it further from the pole (origin).

Step 3: Observing the Effect of d

With \(e=1\), all resulting shapes are parabolas. The parameter \(d\) affects the size of the parabola, stretching it radially outwards or inwards as \(d\) increases or decreases, respectively.

Step 4: Graphing for d=1, Varying e

Set \(d=1\) and vary \(e\): - When \(e=0.5\), plot \( r = \frac{0.5}{1 + 0.5\sin\theta} \), indicating an ellipse.- When \(e=1\), plot \( r = \frac{1}{1 + \sin\theta} \), indicating a parabola.- When \(e=1.5\), plot \( r = \frac{1.5}{1 + 1.5\sin\theta} \), indicating a hyperbola.Observe how the shape changes from ellipse to parabola to hyperbola as \(e\) increases.

Step 5: Observing the Effect of e

The parameter \(e\) (eccentricity) determines the type of conic:- \(0 < e < 1\) results in an ellipse.- \(e = 1\) results in a parabola.- \(e > 1\) results in a hyperbola. As \(e\) increases from 0 to more than 1, the conic opens up from an ellipse to a hyperbola.

Key Concepts

Eccentricity

Eccentricity is a key factor in determining the shape of conic sections in polar coordinates. In simple terms, eccentricity, denoted by the letter \(e\), is a measure of how "stretched" or "circulated" a conic section is.

• If the eccentricity is greater than 0 but less than 1, the conic is an ellipse. An ellipse looks like a flattened circle and has two foci.
• An eccentricity of exactly 1 forms a parabola. Parabolas have one focus and are symmetric U-shaped curves.
• Eccentricity greater than 1 results in a hyperbola, consisting of two mirrored open curves.

As the eccentricity increases from 0 toward 1, the ellipse becomes more elongated. When eccentricity approaches or equals 1, the curve becomes a parabola, opening wider until it forms a hyperbola if it exceeds 1. The value of eccentricity fundamentally affects the path and appearance of the conic section.

Semi-latus Rectum

The semi-latus rectum is a portion of the conic section that plays a crucial role in understanding its geometry. It is denoted by \(d\) in polar equations and represents the perpendicular distance from a focus to the directrix.
In the given equation, the semi-latus rectum helps define the size or scale of the conic. Depending on its value:

• Changing \(d\) when \(e=1\) (a parabola) affects the radial distance of the parabola from the origin. Simply put, larger values of \(d\) stretch the parabola outward, while smaller values make it closer to the origin.

Semi-latus rectum offers insights into the "spread" of the conic regarding its width and the proximity to the center in polar coordinates. By manipulating \(d\), you can visualize different sizes and placements of the same conic section without altering its fundamental shape.

Types of Conic Sections

Conic sections refer to the curves obtained from the intersection of a plane with a cone. These include ellipse, parabola, and hyperbola, each distinguished by their unique properties and applications.

• Ellipses occur when the plane cuts through both nappes of the cone but does not pass through its base. They closely resemble ovals and are common in planetary orbits.
• Parabolas appear when a plane is parallel to the slant height of the cone. These curves are notably found in satellite dishes and car headlights due to their reflective properties.
• Hyperbolas result when the intersecting plane passes through both nappes, creating two mirrored curves. Hyperbolas are prevalent in navigation systems and certain architectural designs.

Understanding these types helps us predict the motion or pathway of objects that follow these curves. Whether it's the orbit of a planet or the path of a trajectory, conic sections provide a framework for analyzing their movement.