Problem 42 - Algebra and Trigonometry

Question

(a) Graph the conics $$ r=\frac{e d}{(1+e \sin \theta)} $$ 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

The value of \( d \) affects the size of the parabola, not its shape. The value of \( e \) determines the type of conic section: ellipse, parabola, or hyperbola, and significantly changes the conic's shape.

Step 1: Understand the Equation

The given polar equation is \( r = \frac{e d}{1 + e \sin \theta} \). This equation represents a conic section in polar coordinates. The parameter \( e \) is the eccentricity, which determines the type of conic: ellipse (0 < e < 1), parabola (e = 1), or hyperbola (e > 1). The parameter \( d \) is related to the semi-latus rectum of the conic.

Step 2: Analyze the Effect of \(d\) with \(e=1\)

For \( e = 1 \), the conic is a parabola. Substitute \( e=1 \) into the equation: \( r = \frac{d}{1 + \sin \theta} \). Graph this equation for various values of \( d \) such as 1, 2, 3, etc. As \( d \) increases, the parabola shifts outward, but its shape remains the same because \( d \) affects the size, not the shape.

Step 3: Graph Parabolas with Various \(d\)

Graph the polar equations for \( d = 1, 2, 3, \ldots \). You will notice that larger values of \( d \) result in a larger parabola, but the general shape of the parabola does not change. Thus, \( d \) scales the size of the parabola without affecting its curvature.

Step 4: Analyze the Effect of \(e\) with \(d=1\)

For \( d = 1 \), substitute \( d=1 \) into the equation to get \( r = \frac{e}{1 + e \sin \theta} \). Now, graph this equation for various values of \( e \) like 0.5, 1, and 1.5 to see different conics: 0.5 (ellipse), 1 (parabola), 1.5 (hyperbola). The eccentricity \( e \) determines the type of conic and its shape.

Step 5: Graph Conics with Various \(e\)

Graph the polar equations for \( e = 0.5, 1, 1.5, \ldots \). If \( e < 1 \), we get ellipses that become more circular as \( e \) approaches zero. At \( e = 1 \), the graph is a parabola, and for \( e > 1 \), the graph becomes a hyperbola. \( e \) has a critical impact on the type and shape of the conic.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when talking about conic sections. It is often denoted by the letter \( e \). This parameter helps us understand what kind of conic section we are dealing with and gives us clues about its shape.
The three main types of conic sections are:

• Ellipse: Occurs when \( 0 < e < 1 \). The shape is oval, and the eccentricity measures how "stretched" the ellipse is. As \( e \) approaches 0, the ellipse becomes more circular.
• Parabola: Formed when \( e = 1 \). Parabolas are U-shaped curves, and in this case, they represent a conic section that opens outward and never closes.
• Hyperbola: Happens when \( e > 1 \). Hyperbolas have two separate curved branches that open in opposite directions.

In the context of our original equation, varying the value of \( e \) fundamentally alters the nature of the graph. A greater understanding of eccentricity aids in predicting and visualizing different conic shapes.

Polar Coordinates

Polar coordinates are crucial when graphing conic sections, especially when the equation is given in polar form. They describe a point in the plane by two values: the radial distance \( r \) from a fixed point, known as the pole, and the angle \( \theta \), which is measured from a fixed direction. This system is particularly useful in situations where one deals with curves like the conics, which can have rotational symmetry.
The original exercise provided an equation in polar coordinates: \( r = \frac{e d}{1 + e \sin \theta} \). Here, the role of \( r \) is to define how far the point is from the pole, while \( \theta \) helps in determining its direction.
Utilizing polar coordinates allows for a more intuitive understanding and visualization of conic sections, particularly when these shapes are centered around a single point. Additionally, this coordinate system can simplify the mathematics of complex curves.

Graphing Conics

When graphing conic sections, it is essential to identify how changes in parameters such as the eccentricity \( e \) and the other parameters impact their overall shape. In our case, these graphs are plotted using polar coordinates which provide a robust way of visualizing them.

• By varying \( d \) with \( e = 1 \), we observe that conics transform into parabolas that scale in size but retain shape. This is because the semi-latus rectum, linked with \( d \), affects only the dimension and not the curvature.
• On the other hand, varying \( e \) with \( d = 1 \) reveals much about the conic's identity itself. For instance, \( e < 1 \) shows ellipses, \( e = 1 \) results in a parabola, and \( e > 1 \) leads to hyperbolas.

Observing these transformations on a graph gives vital insights into the variety and adaptability of conic structures. Understanding how to graph and interpret these changes can significantly enhance one's grasp of the geometric properties inherent to conic sections.