Problem 6 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the polar equation for (a) \(e=1,\) (b) \(e=0.5,\) and \((\mathrm{c}) e=1.5 .\) Identify the conic for each equation. $$r=\frac{2 e}{1-e \cos \theta}$$

Step-by-Step Solution

Answer

The conic section for \(e = 1\) is a parabola, for \(e = 0.5\) it's an ellipse and for \(e = 1.5\) it's a hyperbola.

Step 1: Graph for e=1

Use a graphing utility and graph the polar equation \(r = \frac{2*1}{1 - 1*\cos \theta} = \frac{2}{1 - \cos \theta}\) . As the value of \(e\) is one for this case, the graph will represent a conic section that corresponds to \(e = 1\).

Step 2: Graph for e=0.5

Graph the polar equation \(r = \frac{2*0.5}{1 - 0.5*\cos \theta} = \frac{1}{1 - 0.5*\cos \theta}\). Now, the eccentricity is 0.5, which will give a different conic section.

Step 3: Graph for e=1.5

Finally, graph the polar equation \(r = \frac{2*1.5}{1 - 1.5*\cos \theta} = \frac{3}{1 - 1.5*\cos \theta}\). Yet again, the change in eccentricity (now, \(e = 1.5\)) will result in another distinctive conic section.

Step 4: Identification of the Conic sections

Now, identify the conic sections for each graph. If the graph is a U-shaped curve, it's a parabola (\(e = 1\)). If it looks like an elongated circle (oval), it's an ellipse (\(e < 1\)). If it consists of two separate curved lines, it's a hyperbola (\(e > 1\)).

Key Concepts

Conic Sections in Polar Equations

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These curves include ellipses, parabolas, and hyperbolas. A polar equation of the form \[r = \frac{2e}{1 - e\cos\theta}\]represents these conic sections in a polar coordinate system. By adjusting the eccentricity, denoted as \(e\), we can observe different conic sections:

• For \(e = 1\), the conic section is a parabola. It creates a U-shaped curve on the graph.
• When \(e < 1\), such as 0.5, it forms an ellipse. It's typically oval-shaped, resembling a stretched circle.
• If \(e > 1\), like 1.5, it results in a hyperbola, identified by its distinct pair of curves.

Understanding these basic shapes helps in recognizing the conic sections by examining their graphs. Beginners may visualize these types using a graphing utility, which is crucial for understanding polar equations.

Eccentricity of Conics

Eccentricity is a number that describes the shape of a conic section. It specifically defines how much the conic section deviates from being circular. Here's a breakdown of eccentricity values for different conics:

• An ellipse has an eccentricity less than 1 (\(e < 1\)). The smaller the eccentricity, the more the ellipse approximates a circle.
• A parabola has an eccentricity equal to 1 (\(e = 1\)). It reflects a balance between an open curve and bounded shape.
• Hyperbolas possess an eccentricity greater than 1 (\(e > 1\)), exhibiting two separate branches.

Recognizing the eccentricity is crucial for identifying and distinguishing various conic sections. By using this attribute, one can instantly infer the nature of the conic section just by checking the eccentricity value of the polar equation. This aspect greatly aids in simplifying complex polar equations.

Graphing Utility for Polar Equations

Graphing utilities, like calculators and computer software, can assist in visualizing polar equations. For equations in the form \[r = \frac{2e}{1 - e\cos\theta}\]such tools are extremely valuable.Here's how a graphing utility helps:

• It enables the visualization of conic sections by plotting the polar equations directly, allowing for better understanding and analysis.
• Adjusting the value of \(e\) on the utility reflects immediate changes in the graph, showing real-time transformations from one conic section to another.
• The utility identifies intersections, vertexes, and directrices, which are hard to sketch manually, accurately.

Such resources are invaluable for students challenged by polar coordinates, as they transform abstract mathematical concepts into concrete visual representations. Developing familiarity with these utilities not only assists in homework but also replicates in professional mathematical work.