Problem 291 - Calculus Volume 3

Question

Sketch the graph of each conic. $$ r=\frac{4}{1+\cos \theta} $$

Step-by-Step Solution

Answer

The graph is a parabola opening to the right with vertex at \((2,0)\).

Step 1: Identify the Conic Type

The formula provided, \(r=\frac{4}{1+\cos \theta}\), resembles the standard polar form of a conic, \(r = \frac{ed}{1 + e\cos \theta}\), where the denominator has a term involving \(\cos \theta\). This suggests that the conic is either an ellipse or a hyperbola. Since \(e > 0\) and it corresponds to the conic \(e = 1\), this specific equation represents a parabola.

Step 2: Determine Orientation and Directrix

The conic is in polar coordinates, and the form \(1 + \cos \theta\) places the directrix horizontally. Specifically, \(\cos\theta\) indicates rightward or leftward orientation. For \(e = 1\) and the positive sign before \(\cos\theta\), the parabola is directed towards the right.

Step 3: Calculate the Vertex

For a parabola with \(r = \frac{4}{1 + \cos \theta}\), the vertex directly corresponds with \(\theta = 0\). Calculate it by substituting \(\theta = 0\) into the equation: \(r = \frac{4}{1 + \cos 0} = \frac{4}{2} = 2\). Hence, the vertex in polar coordinates is \((2, 0)\).

Step 4: Graph the Parabola

Convert key points into Cartesian coordinates to assist in plotting. The vertex \((2, 0)\) translates to \((2, 0)\) in Cartesian coordinates. Sketch the graph with the vertex at this position and opening to the right. Ensure the directrix, which runs vertically through \(x = -2\), is shown for reference, although the graph extends infinitely.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe points in the plane using a combination of radial distances and angular directions. Think of moving from the center of a circle outwards by a certain distance, and then rotating around a central point. This is different from Cartesian coordinates, which rely on horizontal and vertical distances along the x and y axes, respectively.

In polar coordinates, each point is represented by \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle from the positive x-axis. Angles are typically measured in radians.

• If \(r\) is positive, the point is in the direction of the angle \(\theta\).
• If \(r\) is negative, the point is in the opposite direction of \(\theta\).

Understanding polar coordinates is essential when dealing with conic sections in polar form, as seen in the equation \(r=\frac{4}{1+\cos \theta}\). This representation is beneficial for problems that have a radial symmetry or are best described in terms of angles.

Parabolas

A parabola is a specific type of conic section that is defined by a set of points equidistant from a line called the directrix and a fixed point called the focus. Its shape is a symmetric U or inverted U, but in polar coordinates or other orientations, it might appear in different orientations.

For the given equation \(r=\frac{4}{1+\cos \theta}\), understanding that it represents a parabola depends on recognizing the conic form and the key parameter \(e\) (eccentricity). When the eccentricity \(e = 1\), the conic section is a parabola.

Remember:

• The focus of a parabola in polar coordinates can be at the origin, affecting how \(\theta\) and \(r\) interact.
• Orientation in polar form can direct the graph sideways, upwards, or downwards, depending on if \(\cos \theta\) or \(\sin \theta\) is present.

This specific parabola equation points to a rightward opening, signifying its focus lies along the direct cartesian line derived from \(\theta = 0\).

Graphing Conics

Graphing conic sections like our parabola using polar coordinates can be challenging but rewarding. Begin by identifying key points and how the conic is oriented. The directrix and vertex are crucial for outlining the shape.

For our example \(r=\frac{4}{1+\cos \theta}\):

• The vertex is found by evaluating at \(\theta = 0\), resulting in the point \( (2, 0) \) in polar coordinates.
• In Cartesian coordinates, this translates directly to \( (2, 0) \).

To accurately sketch conics, consider a handful of points over typical \(\theta\) values, ensuring a smooth curve. Observing these key features helps in plotting more complex shapes, whether they are open-ended parabolas or closed ellipses and circles.
Visualize:

• The direction of the opening of the parabola can help establish how far you need to sketch.
• The role of the directrix is often more subtle in polar graphs but remains vital for understanding symmetry and reference.

With practice and familiarity, graphing conics in polar coordinates develops into a visual and intuitive process.