Problem 69 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(65-74.\) $$r=8 /(4+\cos \theta)$$

Step-by-Step Solution

Answer

Graph the polar equation as an ellipse centered on the origin, reflecting the section of a conic given \( r = \frac{8}{4+\cos\theta} \). Compare its plot to traditional ellipses for verification.

Step 1: Identify the form of the equation

The equation given is in polar coordinates, specifically the form of a conic section with the equation \( r = \frac{k}{1 + e \cos \theta} \). By comparing these, we match \( \, \frac{k}{4 + \cos \theta} = \frac{k}{1 + e \cos \theta} \, \) to identify potential values for \( e \) and \( k \).

Step 2: Rewrite the equation

Notice the equation is written as \( r = \frac{8}{4 + \cos \theta} \). Rewrite it as \( r = \frac{8}{1+ \frac{1}{4} \cos \theta} \). This helps in identifying it as the standard form for conic sections with the constant term and a cosine term.

Step 3: Identify parameters

By rewriting the polar equation, we realize that the conic section is a hyperbola since the standard form shows the addition involving \( \cos \theta \, \). More precisely, it resembles the expression where the numerator indicates \( k \) and comparing with the denominator part identifies \( e = \frac{1}{4} \).

Step 4: Graph the conic section

In polar coordinates, when \( r = \frac{k}{1 + e \cos \theta} \) and \( e = \frac{1}{4} \), the directrix can be determined, which is necessary in drawing conic sections. However, \( e \leq 1 \) implies an ellipse. Plotting the graph involves determining unique points by substituting \( \theta \) values. Calculate critical values (\( \theta = 0, \pi/2, \pi, 3\pi/2 \)) and use symmetry to complete the graph.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to locate points in a plane compared to the traditional Cartesian coordinates. Instead of using an x-y grid, polar coordinates utilize a combination of an angle and a radius. This means each point is described by \( (r, \theta) \), where \( r \) indicates how far away the point is from the origin and \( \theta \) is the angle from the positive x-axis.

This system is particularly useful for problems involving circular or rotational symmetry. For example, conic sections like circles, ellipses, and hyperbolas can be described elegantly using polar equations, where variations of \( r = \frac{k}{1 + e \cos \theta} \) come into play.

• Beneficial for plotting curves that are naturally circular.
• Good for graphing situations where measuring angles and distances from a central point is essential.

Understanding polar coordinates is essential for anyone delving into more complex mathematical problems involving rotational systems or conics.

Graphing Conic Sections

Graphing conic sections is a unique skill that involves understanding how certain algebraic equations translate into visual shapes such as ellipses, hyperbolas, and parabolas. These shapes are derived from the intersection of a plane with a cone.

In the context of polar coordinates, these sections are often represented by equations like \( r = \frac{k}{1 + e \cos \theta} \). Here, identifying the parameters, \( k \) and \( e \), is crucial, as they dictate the shape and size of the conic:

• When \( e = 0 \), the conic is a circle, as the distance is constantly defined.
• When \( 0 < e < 1 \), the conic is an ellipse.
• When \( e = 1 \), it forms a parabola.
• When \( e > 1 \), it results in a hyperbola.

Plotting these involves calculating critical points by substituting values for \( \theta \) and drawing the shape based on symmetry and specific properties defined by \( e \) and \( k \). It's a satisfying process that links algebra and geometry.

Hyperbolas

Hyperbolas are one of the intriguing shapes classified as conic sections. Unlike ellipses, hyperbolas are defined as the set of all points such that the absolute difference of the distances to two fixed points (foci) is constant. They consist of two separate branches- each a mirror image of the other.

Within polar coordinates, a hyperbola can be described by an equation of the form \( r = \frac{k}{1 + e \cos \theta} \) where \( e > 1 \). This signifies the eccentricity of the conic section, revealing how elongated the curve is. Understanding hyperbolas involves knowing certain features:

• Twin symmetrical branches extending towards infinity.
• The center is not part of the curve but serves as a symmetry axis.
• The focus lies along the transverse axis.

Graphing a hyperbola requires using specific points and understanding its characteristic shape, identifying the directrix, and plotting based on symmetry derives from \( e \) and the form of the equation.

Ellipse Identification

Identifying ellipses from conic section equations involves understanding the parameter \( e \), the eccentricity. An ellipse is a conic where \( 0 < e < 1 \). In polar coordinates, this can sometimes be less evident, so recognizing the basic form of the equation is crucial.

The standard polar form \( r = \frac{k}{1 + e \cos \theta} \) assists in this recognition. If the value of \( e \) satisfies the condition \( 0 < e < 1 \), the graph will portray an ellipse.

Ellipses have several hallmark characteristics:

• A smooth, closed, and oval shape.
• Defined by two foci and a constant sum of distances from any point on the ellipse to these foci.
• The center of an ellipse in polar coordinates coincides with the origin.

Understanding these features helps distinguish ellipses from similar conics like circles or hyperbolas, allowing you to visualize and label them appropriately.