Problem 27 - Contemporary Precalculus

Question

Sketch the graph of the equation and label the vertices. $$r=\frac{15}{3-2 \cos \theta}$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the polar equation \(r = \frac{15}{3-2\cos \theta}\) and identify its vertices. Answer: The graph of the given polar equation is an ellipse with Cartesian form \(x^2 + y^2 = \frac{15}{1-\frac{2x}{3}}\). To find the vertices, set the derivative with respect to \(\theta\) equal to 0 and solve for \(\theta\). Then, plug these values into the original polar equation to obtain the corresponding vertices. Once the vertices are identified, sketch the ellipse by connecting them and label the vertices on the graph.

Step 1: Identify the shape of the curve

First, recognize that the polar equation is in the form of a conic with a denominator in the form \((1-e\cos \theta)\), where \(e\) is the eccentricity. When \(e = 0\), the equation represents a circle. When \(0 < e < 1\), the equation represents an ellipse. In our case, \(0 < e = 2 < 1\), indicating that the graph represents an ellipse.

Step 2: Identify the equation in Cartesian coordinates

Convert the polar equation to Cartesian coordinates using the substitution \(x = r\cos \theta\) and \(y = r\sin \theta\): $$r = \frac{15}{3-2\cos \theta}$$ Multiply by the denominator (\((3 - 2\cos \theta)\)) to eliminate the fraction: $$r(3 - 2\cos \theta) = 15$$ Now substitute the Cartesian coordinate form for \(r\) and \(\cos \theta\): $$(x^2+y^2)(3 - 2\frac{x}{x^2+y^2}) = 15$$ Simplify the equation: $$x^2 + y^2 = \frac{15}{1-\frac{2x}{3}}$$ This is the Cartesian form of the equation, which is more easily plotted.

Step 3: Find the vertices

The vertices of the ellipse are the points where the curve is furthest from the center. We can find them by setting the derivative with respect to \(\theta\) equal to 0 and solving for \(\theta\): $$\frac{d}{d\theta} \left(\frac{15}{3-2\cos \theta}\right) = 0$$ This will give us two values for \(\theta\), and plugging them back into the original polar equation will give us the corresponding vertices of the ellipse.

Step 4: Sketch the graph and label the vertices

Plot the Cartesian coordinate equation with the vertices found in the previous step. Then, sketch the ellipse by connecting the vertices and ensuring the shape matches the identified ellipse from Step 1. Finally, label the vertices on the graph. With these steps completed, the graph of the polar equation should be accurately represented with the vertices labeled.

Key Concepts

Conic Sections

Conic sections are fundamental shapes obtained through the intersection of a plane with a double cone. These include circles, ellipses, parabolas, and hyperbolas. Understanding conic sections is essential as they often appear in mathematical descriptions of natural phenomena.

• A **circle** arises when the intersecting plane is perpendicular to the axis of the cone. It is the simplest conic section.
• An **ellipse** is formed when the plane intersects at an angle, producing an elongated circle.
• A **parabola** results when the plane is parallel to one of the slopes of the cone.
• A **hyperbola** forms when the plane intersects both halves of the cone.

The shape of the conic section obtained depends on the angle and distance relative to the cone's axis. Polar coordinates are frequently used to describe conics, particularly when they are not centered at the origin or have complex orientations.

Ellipse

An ellipse is a conic section characterized by its oval shape, appearing as a stretched out circle. It is defined as the set of points where the sum of the distances from two fixed points, called foci, is constant.
What makes an ellipse unique among conic sections are its axes:

• The **major axis** is the longest diameter of the ellipse, passing through both foci.
• The **minor axis** is perpendicular to the major axis, intersecting it at the center of the ellipse.

In polar coordinates, the equation for an ellipse often appears with a denominator that includes cosine or sine functions, indicating its relation to angles in the plane. The shape and orientation of the ellipse depend on these angular terms and coefficients.
In the presented exercise, the given polar equation represents an ellipse, indicating its elongated circular form with specific major and minor axes derived from solving the provided equations.

Graph Sketching

Graph sketching involves visualizing equations on a coordinate system, facilitating a visual understanding of mathematical relationships. For students, sketching helps in grasping the symmetry, behavior, and properties of equations.
When sketching for an ellipse in polar coordinates, follow these essential steps:

• **Convert Polar to Cartesian**: It may be easier to visualize a graph once the polar equation is expressed in Cartesian coordinates.
• **Identify Key Features**: Locate vertices, center, major and minor axes, and use these to plot important points of the ellipse.
• **Plot and Connect**: Begin plotting with the vertices and extend the curve, ensuring it aligns with the identified shape—an elongated circle in this case.

This practice strengthens comprehension of the equation and conic properties, aiding in accurate representation of elliptical shapes on graphs.

Eccentricity

Eccentricity is a numerical measure that describes how much a conic section deviates from being circular. It is one of the most important properties that categorize the specific type of conic.
For ellipses, eccentricity ( \( e \)) ranges from 0 to 1, where:

• \(e = 0\) **represents a circle**, meaning no deviation, and the ellipse takes a perfect circular form.
• \(0 < e < 1\) **indicates an ellipse with varying degrees of stretching**: the closer \(e\) is to 1, the more elongated the ellipse.

The given exercise showcases an eccentricity value that confirms the equation represents an ellipse. Understanding eccentricity helps predict and describe the ellipse's shape before sketching—it clarifies how the foci and distances affect the overall form, aiding in accurate graphical representation.