Problem 42 - Precalculus: Mathematics for Calculus

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

Changing \( d \) modifies the parabola size, and changing \( e \) changes the conic type and shape.

Step 1: Understanding the Conic Section Equation

The given polar equation is \( r = \frac{e d}{1 + e \sin \theta} \), which represents a conic section in polar coordinates. The parameter \( e \) is the eccentricity, determining the type of conic, and \( d \) is the directrix, influencing its scale.

Step 2: Graphing for Fixed Eccentricity (e = 1)

For \( e = 1 \), the conic is a parabola. Vary \( d \) (e.g., \( d = 0.5, 1, 2 \)) and graph each conic. A smaller \( d \) results in a smaller-scale parabola, while a larger \( d \) results in a larger, more stretched parabola.

Step 3: Effect of Parameter 'd'

The parameter \( d \) acts as a scaling factor. As \( d \) increases, the parabola widens, and as \( d \) decreases, it narrows. Thus, \( d \) affects the size of the conic section but not its type when \( e = 1 \).

Step 4: Graphing for Fixed Directrix (d = 1)

Fix \( d = 1 \) and vary \( e \). For \( e = 0.5, 1, 1.5 \), graph the conics. This will show different types of conics: \( e < 1 \) (an ellipse), \( e = 1 \) (a parabola), and \( e > 1 \) (a hyperbola).

Step 5: Effect of Eccentricity 'e'

The eccentricity \( e \) determines the type of conic: an ellipse if \( e < 1 \), a parabola if \( e = 1 \), and a hyperbola if \( e > 1 \). As \( e \) varies from less than 1 to greater than 1, the conic transitions through these types.

Key Concepts

Eccentricity

Eccentricity, denoted by the symbol \( e \), is a key parameter in defining the shape of a conic section. It tells us how much the conic deviates from being circular. Depending on its value, the conic section can take different forms:

• **Ellipse**: If \( e < 1 \), the conic is an ellipse, which means the shape looks like an elongated circle. The smaller the \( e \), the more rounded the ellipse appears.


• **Parabola**: If \( e = 1 \), the conic becomes a parabola. Parabolas are mirror-symmetrical and open shapes.


• **Hyperbola**: If \( e > 1 \), the conic is identified as a hyperbola. Hyperbolas look like two opposite-facing curves.

In essence, eccentricity serves as a measure of how "stretched" a conic section is. The transition from ellipse to parabola to hyperbola is smooth as \( e \) increases. By merely adjusting this parameter, we can completely change the nature of the conic in question. Understanding eccentricity helps in predicting the conic's geometric layout.

Polar Coordinates

Polar coordinates are a way to define the position of a point in a two-dimensional plane. Unlike the Cartesian coordinate system, which uses \( x \) and \( y \) coordinates, polar coordinates use the distance from a reference point and an angle from a reference direction to specify locations.

• The reference point, often labeled as the origin or pole, serves as the starting point for measure.


• The angular coordinate, \( \theta \), is the angle measured from a fixed direction, commonly the positive x-axis.


• The radial coordinate, \( r \), is the direct distance from the pole to the point.

Conic sections can be represented in polar form, as seen in the equation \( r = \frac{e d}{1 + e \sin \theta} \). Here, \( r \) relates directly to the distance from the origin to a point on the conic curve. This method simplifies graphing some curves, especially when symmetry around a reference axis is involved. Understanding polar coordinates allows for a more natural description and deduction of properties related to circular and conic pathways.

Directrix

The directrix is another fundamental component in understanding conic sections. It is a straight line that helps in defining the conic along with a focus and eccentricity. Essentially, the directrix serves as a reference line that, together with the focus, helps describe the set of points that form the conic section.

• For ellipses and parabolas, there is usually one directrix, while hyperbolas have two.


• The role of the directrix in equations is crucial, as seen in \( r = \frac{e d}{1 + e \sin \theta} \). The parameter \( d \) corresponds to the distance from a directrix.


• When dealing with parabolas, the distance from any point on the parabola to the focus equals the perpendicular distance from the point to the directrix.

The concept of a directrix helps in visualizing and plotting the conic section in relation to the focus and other geometrical properties. By adjusting \( d \), as noted in the exercise, we can alter the scale of the conic while preserving its type when \( e = 1 \). Therefore, understanding the directrix offers a clearer insight into the geometrically constructed definitions of conics.