Problem 29 - Precalculus

Question

Use a graphing utility to graph the polar equation. Identify the graph. $$r=\frac{3}{-4+2 \cos \theta}$$

Step-by-Step Solution

Answer

The graph of the given polar equation \(r = \frac{3}{-4 + 2\cos \theta}\) is an ellipse.

Step 1: Understand the polar equation

Recognize the form \(r = \frac{a}{1 + e \cos \theta}\). This is a form of a conic section with eccentricity e. The equation can be rewritten in the form \(r = \frac{3}{1 + 2 (\cos \theta - 2)}\). Here, \(a = 3\) and \(e = -2\). Since \(e < 1\), the graph represents an ellipse.

Step 2: Graph the polar equation

Plot the polar equation using a graphing utility. Ensure that the graphing utility is set to 'polar mode'. Plot several points for a range of values of \(\theta\) and connect these points smoothly to form a curve. Ensure that you plot enough points to capture the essence of the curve.

Step 3: Identify the graph

Based on the eccentricity determined in the first step and the final graph obtained in the second step, the resultant shape should be identified. For eccentricity \(e < 1\), we should have an ellipse.

Key Concepts

Eccentricity

Eccentricity is a fundamental aspect of conic sections. It helps us understand the shape and nature of the curve we are analyzing. To put it simply, eccentricity is a measure of how "un-circular" a conic section is. It directly relates to how a conic stretches and squashes.
The eccentricity (\( e \) in mathematical terms) is crucial for identifying different conic sections:

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

In the context of polar equations such as \( r = \frac{3}{-4 + 2 \cos \theta} \), we identify:\( e = -2 \). Although typically negative eccentricities are not standard, this happens due to the way polar equations are rearranged. In practice for this problem, you treat it as if \( e = 2 \). Since it's absolute value \( (e = 2) \) is greater than 1, it suggests we are looking at related theories in polar graphs hence more adjustments are needed.
The negative sign might hint at an orientation factor rather than shape.

Conic Sections

Conic sections are a set of curves obtained by slicing a cone with a plane. The main types of conic sections include circles, ellipses, parabolas, and hyperbolas.
Each of these shapes has a specific set of characteristics:

• Circle: Perfectly round and symmetrical.
• Ellipse: Looks like a flattened or elongated circle.
• Parabola: A unique open curve, often seen in reflective surfaces.
• Hyperbola: Consists of two separate curves mirrored against each other.

The form of a conic section in polar coordinates can appear as \( r = \frac{a}{1 + e \cos \theta} \). Here, \( a \) is a constant, and \( e \) is the eccentricity which we discussed earlier. Based on the eccentricity, the curve's nature can be determined.
In our exercise, despite how the equation appears, the task is to identify the graphical representation. Due to the expression used, it is defined as a form recognizable by graphing utilities. The key is noting eccentricity and rearranging the expression to evaluate it accurately within the conic forms.

Graphing Utilities

Graphing utilities are incredibly useful tools for visualizing mathematical concepts like polar equations. They help us plot complex equations and identify curves such as those formed by conic sections.
When dealing with polar equations, like the one in our exercise, graphing utilities can simplify how we see the dynamics of the equation:

• Polar Mode: Ensure your graphing utility is set to polar mode.
• Plotting Points: Graph several points by varying \( \theta \) to see the overall shape.
• Curves Visualisation: Connect the plotted points smoothly, as this demonstrates the true form of the equation.
• Understanding Shapes: Compare your graph to theoretical recognitions to understand the nature (circle, ellipse, parabola, hyperbola).

Graphing utilities make it much easier to connect the mathematical formulas to visual forms. This experience bridges the gap between abstract equations and tangible visuals, enticing students to explore more dynamic mathematical frameworks.
In solving the original exercise, the graphing utility was crucial to realizing that our conic section was indeed an ellipse - not immediately obvious just from the equation itself.