Problem 26 - 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

Varying d scales the conic's size for parabolas; varying e changes the conic type.

Step 1: Understanding the Conic Equation

The given conic section equation is in the polar form \( r = \frac{ed}{1 + e \sin \theta} \), where \( e \) is the eccentricity and \( d \) is the semi-latus rectum. Changing these parameters will influence the shape of the conic.

Step 2: Case (a): Varying d with e=1

For \( e = 1 \), the conic is a parabola. By varying \( d \), the distance from the focus to the directrix changes. Graph \( r = \frac{1 \cdot d}{1 + 1 \cdot \sin \theta} = \frac{d}{1 + \sin \theta} \) for different values of \( d \). The shape remains a parabola, but its size increases as \( d \) increases.

Step 3: Influence of d in Case (a)

Since \( e = 1 \) results in a parabola, changing \( d \) only affects the scaling of the parabola. A larger \( d \) results in a larger parabola that is more spread out, but the overall shape remains unchanged.

Step 4: Case (b): Varying e with d=1

For \( d = 1 \), graph \( r = \frac{e}{1 + e \sin \theta} \) using different values of \( e \). The value of \( e \) determines the type of conic: if \( e < 1 \), the conic is an ellipse; if \( e = 1 \), it is a parabola; and if \( e > 1 \), it is a hyperbola.

Step 5: Influence of e in Case (b)

By varying \( e \), you actively change the type of conic. Lowering \( e \) below 1 creates an ellipse (almost circular as \( e \to 0 \)), while increasing \( e \) above 1 results in a hyperbola, indicating that \( e \) plays a crucial role in determining the conic type.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when it comes to conic sections. It is denoted by the letter \( e \) and is a measure of how much a conic section deviates from being circular. The value of \( e \) determines the shape of the conic:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), it becomes a hyperbola.

Eccentricity essentially gauges the "stretch" of the conic. As you increase or decrease \( e \), the nature of the conic changes, shifting how wide or narrow it is, and in some cases, changing the conic type entirely. Understanding eccentricity helps in predicting the overall form and properties of the conic.

Polar Coordinates

Polar coordinates offer a different way to represent points on a plane compared to Cartesian coordinates. Instead of using \((x, y)\), polar coordinates use \((r, \theta)\), where:

• \( r \) is the radial distance from the origin to the point.
• \( \theta \) is the angle between the positive x-axis and the line connecting the origin to the point.

In the context of conic sections, equations often use polar coordinates to simplify their representation. For example, the conic section equation \( r = \frac{ed}{1 + e \sin \theta} \) provides insights into the shape of the conic directly through \( e \) and \( \theta \). Converting between Cartesian and polar coordinates can be helpful for solving problems involving conics and allows easier visualization of their geometric properties.

Parabola

A parabola is one of the simpler conic sections and occurs when the eccentricity \( e = 1 \). Parabolas have several notable properties:

• They have a single focus point.
• A directrix line is equidistant from any point on the parabola.
• They are symmetrical with respect to their axis.

In polar form, a parabola retains its shape regardless of changes in the semi-latus rectum \( d \), only its size or scale alters. The equation \( r = \frac{d}{1 + \sin \theta} \) describes parabolas in a way that highlights their vertex and how the latus rectum changes the appearance while preserving the parabolic nature that results in a smooth "U" shape on its graph.

Ellipse

Ellipses occur when the eccentricity \( 0 < e < 1 \). These shapes resemble stretched circles and have several key characteristics:

• Two foci located on the major axis of the ellipse.
• The sum of distances from any point on the ellipse to both foci is constant.
• Ellipses maintain symmetry along both their major and minor axes.

The polar equation for an ellipse, represented when \( e \) is less than 1 in \( r = \frac{ed}{1 + e \sin \theta} \), reveals how changes in \( e \) affect the ellipse's eccentricity. When \( e \) approaches zero, the ellipse becomes more circular. The length of the axes and the positioning of the foci define the specific shape of the ellipse, distinguishing it from other conic forms.

Hyperbola

A hyperbola is another type of conic that occurs when the eccentricity \( e > 1 \). Hyperbolas have distinct features:

• They consist of two separate branches.
• Each branch is mirrored about two axes.
• The difference in distances from any point on a branch to its two foci is constant.

The polar form of a hyperbola, like \( r = \frac{ed}{1 + e \sin \theta} \) with \( e > 1 \), shows how varying \( e \) farther from 1 increases the separation between branches. Understanding hyperbolas is crucial for fields requiring navigation of graphs with distinct parts, such as in physics for waveforms and in compute graphics for rendering separations. Hyperbolas often appear in contexts requiring precision in measurement and representation.