Problem 41 - Precalculus: Mathematics 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

As eccentricity \(e\) increases, the conic becomes more elongated and transitions from an ellipse to a parabola.

Step 1: Understand the Polar Equation of a Conic

The given equation is \( r = \frac{e}{1 - e \cos \theta} \), which represents a conic section in polar coordinates. The shape of the conic depends on the eccentricity \(e\). The conic is a circle if \(e = 0\), an ellipse if \(0 < e < 1\), a parabola if \(e = 1\), and a hyperbola if \(e > 1\).

Step 2: Graph the Conic for e=0.4

Substitute \( e = 0.4 \) into the equation, resulting in \( r = \frac{0.4}{1 - 0.4 \cos \theta} \). Plot this equation using a graphing tool to see the shape of the curve. Since \( e < 1 \), it is an ellipse. The ellipse will appear with a certain magnitude of elongation depending on the value of \(e\).

Step 3: Graph the Conic for e=0.6

Substitute \( e = 0.6 \) into the equation, resulting in \( r = \frac{0.6}{1 - 0.6 \cos \theta} \). Graph this expression to see the resulting curve. The conic is still an ellipse, but the increase in \(e\) makes it appear more elongated compared to \(e = 0.4\).

Step 4: Graph the Conic for e=0.8

Substitute \( e = 0.8 \) into the equation, giving \( r = \frac{0.8}{1 - 0.8 \cos \theta} \). Graph this conic. The shape remains an ellipse, but it becomes even more elongated than for \(e = 0.6\), nearing the shape of a parabola as \(e\) approaches 1.

Step 5: Graph the Conic for e=1

For \( e = 1 \), the equation becomes \( r = \frac{1}{1 - \cos \theta} \). Plot this to visualize the shape. With \( e = 1 \), the conic is a parabola. The curve is no longer closed and opens outward.

Step 6: Compare Shapes of Conics

Compare the shapes obtained from the graphs of \( e = 0.4 \), \(e = 0.6\), \(e = 0.8\), and \(e = 1\). Observe that as \(e\) increases, the conic becomes less circular, more elongated, and eventually changes from an ellipse to a parabola.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when discussing conic sections in mathematics, particularly in polar coordinates. Eccentricity, denoted by the symbol \( e \), is a measurement that describes how much a conic section deviates from being a perfect circle. This value could range from 0 to \( \infty \), and it determines the actual shape of the conic section:

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

Understanding eccentricity helps us predict and visualize the shape and properties of a conic section. It tells us how stretched or open a particular conic will appear in its polar graphical representation.

Ellipse

An ellipse is one of the possible shapes of a conic section characterized by an eccentricity value that falls between 0 and 1. In polar coordinates, this shape is portrayed when the eccentricity \( e \) of the conic section's equation is greater than 0 but less than 1. This implies that the ellipse appears to be an elongated circle:

• The closer \( e \) is to 0, the more circular the ellipse will look.
• As \( e \) approaches 1, the ellipse becomes more elongated or stretched.

Ellipses have two focal points, and any point on the ellipse maintains a constant sum of distances from these two points. This unique property helps define its shape. The graph of an ellipse varies with the change in eccentricity, becoming narrower and more stretched as the value of \( e \) approaches 1 from below.

Parabola

A parabola is a special type of conic section that occurs when the eccentricity \( e \) is precisely 1. In the context of polar coordinates, when a conic section equation exhibits an eccentricity exactly equal to 1, it forms a parabola. This conic does not form a closed shape like an ellipse or circle but instead opens outwards, infinitely extending in one direction:

• The parabola is defined by having a focus and a directrix, with any point on the parabola equidistant from both these parts.
• This unique property leads its characteristic symmetrical and open shape.

Unlike ellipses and circles, the parabola's distinct shape and properties make it useful for many applications, such as in satellite dishes and car headlights, where reflective properties are essential.

Polar Coordinates

Polar coordinates offer a distinct way of representing the location of points in a plane, differing from the typical Cartesian coordinates. In polar coordinates, each point on a plane is specified using a distance from a reference point and an angle from a reference direction. This system suits scenarios like conic sections where curves and their properties are naturally expressed and explored:

• The radial coordinate \( r \) measures the distance from the pole (similar to the origin in Cartesian coordinates).
• The angular coordinate \( \theta \) measures the angle from a reference direction, typically the positive x-axis.

Polar coordinates simplify the representation and graphing of conic sections, such as ellipses, parabolas, and hyperbolas. By using polar equations, mathematicians and scientists can more easily visualize and solve problems involving rotational systems and symmetrical properties.