Problem 3 - College Algebra

Question

If a conic section is written as a polar equation, and the denominator involves sin \(\theta,\) what conclusion can be drawn about the directrix?

Step-by-Step Solution

Answer

The conic section's directrix is vertical.

Step 1: Understand the Polar Equation

A polar equation of a conic section can be expressed in the form \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 - e \sin \theta} \), where \( e \) is the eccentricity and \( d \) is the distance from the pole to the directrix.

Step 2: Identify the Denominator's Part

In these types of equations, the denominator involving sin \( \theta \) indicates an orientation of the directrix. Specifically, when the denominator is of the form involving \( \sin \theta \), the directrix of the conic is vertical.

Step 3: Conclusion About the Directrix Orientation

Since the denominator involves \( \sin \theta \), it is indicative that the directrix of the conic section is parallel to the polar axis. A vertical directrix means it is perpendicular to the horizontal x-axis.

Key Concepts

Eccentricity

Eccentricity, often denoted as \( e \), is a crucial parameter in understanding the shape and nature of conic sections in polar coordinates. It measures the deviation of the conic from being circular. A circle has an eccentricity of 0, making it a perfectly round shape. However, as the eccentricity increases, the conic section becomes more elongated.

In the polar equation for conics, like \( r = \frac{ed}{1 + e \sin \theta} \), the value of \( e \) defines the type of conic section:

• If \( 0 \leq e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

Understanding eccentricity helps in classifying conics and predicting their geometric shapes. It helps students grasp why different values of eccentricity result in completely different types of curves in their mathematical studies.

Directrix Orientation

In the context of polar equations, the orientation of the directrix plays an important role in shaping the conic section. The directrix is a fixed line that, along with a focus, helps to define the conic. The orientation is determined by the trigonometric function present in the denominator of the polar equation.

When the denominator consists of \( \sin \theta \), the directrix is oriented vertically. This means the directrix is perpendicular to the x-axis and does not contribute to any horizontal displacement. Such orientation impacts the way the conic stretches or compresses along the polar plot.

If the denominator were to include \( \cos \theta \), the directrix would then be horizontal, which would influence the conic’s position differently. This understanding of orientation helps solve problems involving polar equations by predicting and visualizing conic sections correctly.

Conic Sections

Conic sections are curves obtained from the intersection of a plane and a cone. They include circles, ellipses, parabolas, and hyperbolas, each having distinctive properties. In polar form, these conic sections are crucial in several mathematical applications like astronomy and physics.

In polar equations such as \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 - e \sin \theta} \), the form of the equation can help identify the type of conic:

• A circle arises when the eccentricity \( e = 0 \).
• An ellipse forms when \( 0 < e < 1 \).
• A parabola occurs when \( e = 1 \).
• A hyperbola appears when \( e > 1 \).

The polar form is particularly useful as it allows these geometric shapes to be expressed through angles and distances from a central point. This idea is foundational in topics beyond pure math, such as modeling orbital paths in physics and engineering, enabling students to draw connections between geometry and practical applications.