Problem 30 - College Algebra

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{6 \csc \theta}{3+2 \csc \theta} $$

Step-by-Step Solution

Answer

The conic is a parabola with directrix \( y = -2 \) and eccentricity \( e = 1 \).

Step 1: Recognize the Conic Form

The given polar equation is \( r = \frac{6 \csc \theta}{3 + 2 \csc \theta} \). A polar equation representing a conic section with the focus at the pole (origin) generally has the form \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 + e\cos\theta} \). Our task is to rearrange it to fit one of these forms and identify the type of conic.

Step 2: Rearrange the Equation

To fit the standard form, convert the equation by factoring \( \csc \theta \) out from both the numerator and the denominator. Start by changing \( \csc \theta \) to \( \frac{1}{\sin \theta} \). This gives: \[ r = \frac{6 \csc \theta}{3 + 2 \csc \theta} = \frac{6/\sin \theta}{3 + 2/\sin \theta} = \frac{6}{3\sin\theta + 2} \] Factor out \( \sin \theta \) from the denominator. This gives: \[ r = \frac{6}{2 + 3\sin\theta} \]

Step 3: Identify the Conic Type

The form \( r = \frac{ed}{1 + e\sin\theta} \) indicates a conic with the focus at the origin when \( e = \) eccentricity. Comparing with \( r = \frac{6}{2 + 3\sin\theta} \), rewrite this as \( r = \frac{6/3}{2/3 + \sin\theta} = \frac{2}{\frac{2}{3} + \sin\theta} \), with the eccentricity being the coefficient of \( \sin\theta \). Set \( e = 1 \).

Step 4: Validate Eccentricity and Identify Directrix

Recall that the eccentricity \( e \) of a conic section corresponds to its type: a circle \( e = 0 \), an ellipse \( 0 < e < 1 \), a parabola \( e = 1 \), or a hyperbola \( e > 1 \). Here, \( e = 1 \), indicating a parabola. For a parabola, the equation has the form \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 - e\sin\theta} \). Simplify: \( r = \frac{2}{1+ \sin\theta} \), forming a parabola with directrix parallel to \( heta = 90^\circ \). The value \( d = 2 \) gives the distance of the directrix from the pole.

Key Concepts

Parabola

A parabola is one of the simplest and most common conic sections. It is formed when a plane cuts through a cone parallel to its side. In the context of polar coordinates, a parabola can be expressed using an equation that includes sine or cosine functions.
The defining property of a parabola is that it has a fixed point known as the "focus" and a directrix, which is a line. Every point on the parabola is equidistant from the focus and the directrix. This balance of distances gives the parabola its smooth, symmetric shape.
It's important to note that in polar coordinates, the equation of a parabola is structured such that its eccentricity \( e = 1 \). This characteristic helps distinguish it from other conic sections—like ellipses and hyperbolas—that have different eccentricity values.
In our exercise, the parabola's focus is at the origin, clearly indicated by the form of the equation. The presence of \( e = 1 \) confirms our conic is indeed a parabola.

Eccentricity

Eccentricity is a key concept in understanding conic sections. It is a numerical measure that describes the shape of a conic section. Every type of conic section has a specific range for eccentricity values:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

Parabolas stand out because they have an eccentricity of exactly 1. This means that the distance from any point on the parabola to the focus is equal to its distance to the directrix. In the exercise, we determined the eccentricity by comparing terms in the rearranged equation, confirming that our conic is a parabola.
Eccentricity not only identifies the conic but also gives insights into the "openness" of a conic. For a parabola, its openness indicates that as one moves along the curve, it gradually diverges away from the vertex but does not close back on itself like a circle or an ellipse.

Directrix

The directrix of a conic section is a straight line used in conjunction with the focus to define and construct the conic. For parabolas, it plays a significant role in conjunction with the focus, where every point on the parabola is equidistant from both the focus and the directrix.
In our exercise, once we identified the equation as a parabola, we found the distance of the directrix from the pole (origin) to be \( d = 2 \). This signifies that the directrix is parallel to the line where \( \theta = 90^\circ \) in polar coordinates.
Understanding the positioning of the directrix relative to the focus helps us visualize the orientation and the exact path of the parabola. It also provides a basis for graphing the conic section accurately, ensuring that all its geometric properties are respected.