Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{8 \csc \theta}{2 \csc \theta-5} $$

Step-by-Step Solution

Answer

The eccentricity is \(\frac{2}{5}\), indicating an ellipse.

Step 1: Recognize the Conic Form

The given polar equation is \( r = \frac{8 \csc \theta}{2 \csc \theta - 5} \). Start by rewriting it in the standard form of a conic. Recall that \( \csc \theta = \frac{1}{\sin \theta} \), thus \( r \sin \theta = y \). Therefore, substitute \( \csc \theta \) with \( \frac{1}{\sin \theta} \).

Step 2: Simplify the Equation

Rewrite the equation by multiplying both the numerator and the denominator by \( \sin \theta \):\[ r \sin \theta = \frac{8}{2 \sin \theta \csc \theta - 5 \sin \theta} \]This simplifies to:\[ y = \frac{8}{2 - 5 \sin \theta} \]Now, the equation describes a conic section in rectangular coordinates.

Step 3: Identify and Classify the Conic

Compare the equation to the general form of a conic \( r = \frac{ed}{1 - e\sin\theta} \). Here, \( e \) is the eccentricity. By rewriting the equation as:\[ r = \frac{8/5}{1 - (2/5)\sin \theta} \]The eccentricity \( e = 2/5 < 1 \), which indicates an ellipse.

Step 4: Determine the Vertices and Sketch

For an ellipse in polar coordinates, the equation \( r = \frac{ed}{1 - e \sin \theta} \) shows vertices at \( \theta = 0 \) and \( \theta = \pi \). Calculate the distances from the focus:For \( \theta = 0 \), \( r = \frac{8/5}{1 - 0} = \frac{8}{5} \).For \( \theta = \pi \), \( r = \frac{8/5}{1 + (2/5)} = 2 \).Sketch the ellipse centered at the origin, marking the vertices.

Key Concepts

Eccentricity of Conics

Eccentricity is a crucial aspect of conic sections, helping to determine their shape and type. In the realm of conic sections, the term eccentricity refers to a number that characterizes how much the conic section deviates from being circular. For an ellipse, which is one of the standard conic sections, this value lies between 0 and 1.

• If the eccentricity ( \( e \)) equals 0, the conic is a perfect circle.
• For an ellipse, the eccentricity is greater than 0 but less than 1.
• If \( e = 1 \), the conic becomes a parabola.
• When \( e > 1 \), the conic transforms into a hyperbola.

In the original exercise, the eccentricity was found to be \( e = \frac{2}{5} \), which is less than 1. This confirms that the conic is an ellipse. Identifying the eccentricity allows us to classify the conic accurately and understand its properties.

Polar Coordinates

Polar coordinates offer an alternative to the standard Cartesian coordinates, particularly useful in problems involving conic sections. Unlike Cartesian coordinates that use horizontal and vertical distances (x, y), polar coordinates involve a radius and angle (\( r, \theta \)). These are excellent for describing shapes that have symmetry around a point, like circles and ellipses.

• The radius \( r \) represents the distance from the origin to the point.
• The angle \( \theta \) is measured from the positive x-axis to the line connecting the origin and the point.

In the exercise, the conic section was described using polar coordinates: \( r= \frac{8 \csc \theta}{2 \csc \theta-5} \). Rewriting it helps us identify the conic form and simplifies sketching its graph. Polar representation offers a clear view of properties such as symmetry and orientation.

Ellipse Properties

Ellipses are a fascinating type of conic section, characterized by their unique shape and distinctive properties. Unlike circles, which have one radius, ellipses have two different foci and a constant sum of distances from any point on the ellipse to these foci.

• Ellipses have two axes: the major axis, which is the longest diameter, and the minor axis, which is the shortest.
• Vertices are the endpoints of the major axis.
• The center of the ellipse is the midpoint between the foci.

In the given problem, the vertices were calculated at polar angles of \( \theta=0 \) and \( \theta=\pi \). The ability to determine vertex locations is crucial in visualizing and sketching an accurate depiction of the ellipse. Understanding these basic ellipse properties allows us to identify and illustrate the conic accurately.