Problem 2 - College Algebra with Corequisite Support

Question

If a conic section is written as a polar equation, what must be true of the denominator?

Step-by-Step Solution

Answer

The denominator must not be zero, so \( e < 1 \).

Step 1: Understanding Polar Equations of Conics

In polar coordinates, conic sections are typically written in the form \( r(\theta) = \frac{ed}{1 + e \cos(\theta)} \) or \( r(\theta) = \frac{ed}{1 + e \sin(\theta)} \), where \( e \) is the eccentricity and \( d \) is a constant.

Step 2: Recognizing the Denominator Structure

The denominator of these polar equations is \( 1 + e \cos(\theta) \) or \( 1 + e \sin(\theta) \). To ensure it corresponds correctly to a conic section, this expression must not equal zero for any \( \theta \), as this would lead to division by zero.

Step 3: Finding Conditions for the Denominator

For the denominator \( 1 + e \cos(\theta) \) or \( 1 + e \sin(\theta) \) not to be zero, the absolute value of \( e \cos(\theta) \) or \( e \sin(\theta) \) must be less than or equal to 1. This means \( e \), the eccentricity, must satisfy the condition \( 0 \leq e < 1 \). This ensures that the conic section is closed, as open conic sections like hyperbolas have different conditions.

Key Concepts

Conic Sections

Conic sections are a fascinating topic in mathematics, involving curves that are created by intersecting a plane with a double-napped cone. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. The type of curve generated depends on the angle and position of the plane intersecting the cone.

• Circle: A circle is formed when the plane is perpendicular to the cone's axis.
• Ellipse: An ellipse results when the angle of the plane is less than that of the cone's slope.
• Parabola: When the plane is parallel to a single slope of the cone, a parabola is formed.
• Hyperbola: A hyperbola occurs when the plane intersects both nappes of the cone.

Understanding conic sections is essential because they are frequently encountered in physics, astronomy, and engineering.
They are used to describe the paths of planets, the design of telescopes, and even the structure of bridges.

Eccentricity

Eccentricity is a measure that helps us understand the "shape" of a conic section. It is denoted by the symbol \(e\) and varies depending on the type of conic:

• Circle: Here, the eccentricity \(e = 0\) as all points are equidistant from the center.
• Ellipse: For an ellipse, \(0 < e < 1\), which indicates how "stretched" the circle is.
• Parabola: In a parabola, \(e = 1\), depicting a curve that extends to infinity.
• Hyperbola: A hyperbola has \(e > 1\), showing even more significant stretching.

The eccentricity tells us how much the conic section deviates from being a perfect circle.
Therefore, knowing \(e\) is crucial for predicting the properties of the conic, like its openness or closeness.

Division by Zero

Division by zero is a mathematical operation that does not result in a defined or finite number. It occurs when you attempt to divide a number by zero, which is undefined and leads to inconsistencies in calculations. In the context of polar equations of conic sections, ensuring that you do not divide by zero is crucial.For instance, in the denominator of the equation \(1 + e \cos(\theta)\) or \(1 + e \sin(\theta)\), if this expression equals zero, you will encounter division by zero. This renders the equation invalid at those points. This typically occurs if the eccentricity \(e\) is such that allows \(\cos(\theta)\) or \(\sin(\theta)\) to neutralize the "1" in the equation.
This is why control over the value of \(e\), ensuring \(0 \leq e < 1\), is vital because it helps maintain a valid conic without undefined behavior.

Polar Coordinates

Polar coordinates provide a unique way to locate points on a plane using a distance and an angle. Unlike Cartesian coordinates that use \((x, y)\), polar coordinates use \((r, \theta)\):

• \(r\): The radius, which is the distance from the origin to the point.
• \(\theta\): The angle formed with the positive x-axis.

This coordinate system is particularly useful for dealing with circular or rotational symmetries, such as in circular motion or waves.Conic sections in polar form become easier to analyze and graph using polar coordinates, as they naturally align with the structure of the conics. Points are calculated through formulas like \(r(\theta) = \frac{ed}{1 + e \cos(\theta)}\).
It becomes easier to understand the shape and orientation of the conic through its polar equation, leveraging the natural attributes of polar coordinates to describe curves.