Problem 25 - Precalculus: Mathematical for Calculus

Question

Graph the conics \(r=e /(1-e \cos \theta)\) with \(e=0.4,0.6,0.8\) and 1.0 on a common screen. How does the value of \(e\) affect the shape of the curve?

Step-by-Step Solution

Answer

Increasing eccentricity \(e\) causes ellipses to elongate; \(e = 1\) forms a parabola.

Step 1: Identify the Conic Type

To identify the type of conic represented by the equation \(r = \frac{e}{1 - e \cos \theta}\), notice that the conic is in polar form. The equation is standard for conics with one focus at the pole. The parameter \(e\) is the eccentricity, which determines the type of conic: \(0 < e < 1\) for ellipses, \(e = 1\) for parabolas, and \( e > 1\) for hyperbolas.

Step 2: Consider Eccentricity Values

For this exercise, we are given specific values of eccentricity: \(e = 0.4, 0.6, 0.8,\) and \(1.0\). We need to graph these conics on a common screen to compare their shapes.

Step 3: Graph the Curve for e = 0.4

When \(e = 0.4\), the conic is an ellipse since \(0.4 < 1\). The graph will show an elongated but closed curve centered around the focus at the pole, with a relatively small deviation from a circle.

Step 4: Graph the Curve for e = 0.6

With \(e = 0.6\), the curve is still an ellipse but more elongated than that of \(e = 0.4\). The graph will show a more noticeable deviation from circularity.

Step 5: Graph the Curve for e = 0.8

For \(e = 0.8\), the ellipse becomes even more pronounced; the elongation increases further, showcasing a significant difference between the major and minor axes.

Step 6: Graph the Curve for e = 1.0

When \(e = 1.0\), the conic becomes a parabola. The graph will show a curve that opens wide and never closes, indicating that the parabola extends infinitely away from the focus.

Step 7: Compare the Shapes

By comparing these graphs, you can observe how increasing the eccentricity \(e\) affects the shape of the conic. For \(0 < e < 1\), as \(e\) increases, the ellipse becomes more elongated. At \(e = 1\), the conic changes to a parabola, reflecting a fundamental shift in the shape.

Key Concepts

Eccentricity

The eccentricity (\(e\)) of a conic section is a non-negative number that defines the shape of the conic. It plays a crucial role in determining the type of conic section you are dealing with:

• For an ellipse, the eccentricity is between 0 and 1 (\(0 < e < 1\)). Lower eccentricity values indicate a shape closer to a circle.
• A parabola has an eccentricity of exactly 1 (\(e = 1\)), resulting in a curve that opens widely and continuously away from its focus.
• If the eccentricity exceeds 1 (\(e > 1\)), the conic takes the form of a hyperbola, characterized by two separate, open branches.

Understanding eccentricity helps in visualizing the nature of conic sections when graphed. The closer the value of \(e\) is to 1, the more stretched or open the curve becomes. This attribute directly impacts the appearance of graphs derived from polar coordinates.

Polar Coordinates

Polar coordinates provide a way to locate points on a plane using a distance and an angle. Unlike the Cartesian coordinate system, which uses horizontal and vertical lines, polar coordinates describe points based on their distance (\(r\)) from a fixed point called the pole (similar to the origin) and an angle (\(\theta\)) from a fixed direction:

• The angle \(\theta\) is typically measured in radians from the positive x-axis.
• The radius \(r\) represents how far the point lies from the pole.
• Points in polar coordinates are expressed as (\(r, \theta\)).

Polar equations can represent various conic sections, utilizing the angle and radius to form curves. This method simplifies calculations for curves around a central point, such as circles and ellipses, which are more cumbersome to express using Cartesian coordinates.

Ellipse

An ellipse in mathematics is a curved shape where any point on the ellipse maintains a constant sum of distances to two focal points. In polar coordinates, an ellipse is characterized by the eccentricity (\(0 < e < 1\)). With its two axes:

• The major axis is the longest diameter, ranging from one side of the ellipse to the other, passing through both foci.
• The minor axis is the shortest diameter, perpendicular to the major axis at the center.

As the eccentricity approaches zero, the ellipse becomes more circular in shape. A higher eccentricity, closer to one, indicates a more elongated ellipse. Given the polar equation \(r = \frac{e}{1 - e \cos \theta}\), it's clear how \(e\) modifies the shape, pressuring the curve to stretch out more as \(e\) nears one. This demonstrates how the ellipse continuously deforms, yet remains bounded, indicating each point on the curve is aligned with the properties mentioned.

Parabola

A parabola is a conic section that appears as a symmetric curve. When the eccentricity \(e\) equals 1 in the polar form equation, the resulting shape is a parabola. Unlike ellipses or circles, parabolas do not close back upon themselves:

• They have a single focal point and a directrix, which is a line equidistant to any point on the parabola relative to its focus.
• The axis of symmetry is a line that passes through the focus and is perpendicular to the directrix, aligning the parabola symmetrically.

In practical use, parabolas are often seen in real-life physics problems, like projectile motion, due to their nature of being open curves extending indefinitely. By analyzing through polar coordinates where \(e = 1\), parabolas are depicted as precise representations of this balanced, enduring curve that does not loop but stretches endlessly, illustrating how graphing in polar serves to easily visualize these properties under such conditions.