Problem 54 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(47-56\) $$ r=1 /(1+\cos \theta) $$

Step-by-Step Solution

Answer

It's a parabola opening to the right.

Step 1: Recognize the polar equation form

The given equation is in the form \( r = \frac{1}{1 + e \cos \theta} \). This is a standard form for a conic section in polar coordinates, where \( e \) is the eccentricity.

Step 2: Identify the conic section type

The equation is in the form of \( r = \frac{1}{1 + \cos \theta} \). Compare it with the standard form \( r = \frac{1}{1 + e \cos \theta} \) to see that \( e = 1 \). When the eccentricity \( e \) is equal to 1, the conic is a parabola.

Step 3: Analyze the parabola's orientation

For the equation \( r = \frac{1}{1+ \cos \theta} \), since the cosine function is involved, the parabola opens to the right, as the cosine term in the denominator relates to the horizontal axis symmetry in polar coordinates.

Step 4: Graph the polar equation

Use polar graph paper or a graphing tool to plot the equation. Start by calculating \( r \) for different values of \( \theta \), particularly focusing on key angles like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and so forth. You'll plot points according to these \( r \) values and then sketch the smooth curve, keeping in mind that this is a parabola opening to the right.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each one has unique properties:

• Circle: All points are equidistant from a center point.
• Ellipse: The sum of distances from any point on the ellipse to two fixed points (foci) is constant.
• Parabola: The distances from any point on the curve to a single point (focus) and a line (directrix) are equal.
• Hyperbola: The difference in distances from any point on the curve to two fixed points (foci) is constant.

When we deal with these sections in polar coordinates, we use equations that describe their shapes based on the eccentricity, denoted as \( e \). The eccentricity determines how "stretched" the conic appears:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

This concept allows us to understand and identify parabolas, such as the one in our exercise, by checking that \( e = 1 \). This feature of polar equations makes it straightforward to classify and graph these intriguing shapes.

Polar Equations

Polar equations are a way of representing curves and shapes using the polar coordinate system. In this system, each point on a plane is determined by a radial distance \( r \) from a fixed origin and an angle \( \theta \) from a fixed direction. This can be especially useful for graphing curves that have symmetry or periodic properties.

A polar equation for a conic section typically takes the form \( r = \frac{1}{1 + e \cos \theta} \) or \( r = \frac{1}{1 + e \sin \theta} \), depending on the axis of symmetry. Here:

• \( e \) is the eccentricity that defines the type of conic.
• \( \cos \theta \) or \( \sin \theta \) determine the orientation of the conic relative to the coordinate axes.

Understanding these forms helps in identifying the kind of conic and its orientation, as polar equations showcase a relationship between \( r \) and \( \theta \) to describe the path of the curve across the polar plane. This variability in angle \( \theta \) enables one to explore all directions from a fixed point, making it a potent tool for visualizing curves like the parabola described in the exercise.

Graphing Techniques

Graphing polar equations involves plotting points in the polar coordinate system and understanding how curves behave relative to the radial distance and angular components. Here are some basic steps for graphing, especially for a parabola using a polar equation:

• Select Key Angles: Focus on angles like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \). Calculate the corresponding \( r \) values for these angles.
• Plot Points: Use the calculated \( r \) values to mark points on polar graph paper. The origin is the center of the paper, and angles are marked counter-clockwise from the positive x-axis.
• Connect Points: Draw a smooth curve through the plotted points, considering the curve’s symmetry and shape.

For equations like \( r = \frac{1}{1+\cos \theta} \), we observe a right-ward opening parabola. Here, understanding the influence of the cosine term is crucial, as it affects the orientation. These graphing techniques allow us to translate complex mathematical relationships into visual representations that are easier to interpret and analyze. Also, utilizing graphing tools or calculators can provide visual feedback, aiding in understanding of the curve's direction and structure. This combination of manual plotting and computational tools makes graphing an interactive and insightful process.