Problem 28 - College Algebra

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(2.5-2.5 \sin \theta)=5 $$

Step-by-Step Solution

Answer

The conic is a parabola with directrix y = 2 and eccentricity 1.

Step 1: Recognize the polar equation format

The given equation is \( r(2.5 - 2.5 \sin \theta) = 5 \). In polar coordinates, this equation is similar to the form \( r = \frac{ed}{1 + e \sin \theta} \), which represents a conic section with the focus at the origin.

Step 2: Simplify to standard form

First, simplify the given equation by distributing the \(r\): \( 2.5r - 2.5r \sin \theta = 5 \). Rearrange to get \( r = \frac{5}{2.5 - 2.5 \sin \theta} \). Factor out 2.5 to rewrite it as \( r = \frac{5/2.5}{1 - \sin \theta} \), simplifying to \( r = \frac{2}{1 - \sin \theta} \).

Step 3: Identify eccentricity and directrix

The standard form \( r = \frac{ed}{1 - e \sin \theta} \) helps identify the conic. In \( r = \frac{2}{1 - \sin \theta} \), the numerator 2 represents \( ed \). Since it's in the form \( 1 - e \sin \theta \), eccentricity \( e = 1 \) for a parabola. Using \( ed = 2 \) and \( e = 1 \), solve \( d = 2 \) to find the parabola's directrix is 2 units away in the positive y-direction.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to locate points on a plane using the distance from a reference point and the angle from a reference direction. Unlike Cartesian coordinates which utilize an x and y-axis, polar coordinates use a point called the pole (similar to the origin) and an angle from a fixed direction, typically the positive x-axis.

This system is particularly useful in situations where symmetry around a point is observed. In polar coordinates, a point is characterized by two values:

• The radial distance, \( r \), which measures how far away the point is from the origin, or pole.
• The angular coordinate, \( \theta \), which expresses the angle from a specified direction.

For example, in the equation given as \( r(2.5 - 2.5 \sin \theta) = 5 \), rearranging it into a standard form helps identify it as a conical section in a polar plane. Polar forms are quite handy when dealing with problems related to spirals, circles, and other conic sections.

Eccentricity

Eccentricity is a crucial concept when studying conic sections as it defines the shape of the conic. It is a non-negative real number that essentially describes how much a conic section deviates from being circular.

There are general ranges of eccentricity values associated with each type of conic:

• For a circle, \( e = 0 \).
• For an ellipse, \( 0 < e < 1 \).
• For a parabola, \( e = 1 \).
• For a hyperbola, \( e > 1 \).

In the problem given, the equation simplifies to \( r = \frac{2}{1 - \sin \theta} \), with \( e = 1 \). This indicates the conic section is a parabola. The clear identification of eccentricity aids in predicting and understanding the inherent propagation and symmetry of the conic section. Understanding eccentricity is key to grasping the fundamental structure and behavior of cones in a polar coordinate system.

Directrix

The directrix is a fixed line associated with a conic section, serving as a determinant line from which distances are measured. It's used alongside the focus to maintain the proportional distance required to define the conic shape.

For parabolas, the distance from any point on the parabola to the focus is equal to the distance from the point to the directrix. That's what gives it the unique property of reflecting light and sound waves to the focus, among other applications.

In the exercise provided, the directrix corresponds to the line 2 units away in the positive y-direction from the pole. This is computed using \( ed = 2 \), where \( e \) and \( d \) represent the eccentricity and the distance to the directrix respectively. Recognizing and strategically using the directrix can often simplify complex proofs and constructions within the realm of conical sections.