Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. \(r=\frac{3}{4-4 \quad \sin \theta}\)

Step-by-Step Solution

Answer

It's a hyperbola with eccentricity 4 and directrix \( d = \frac{3}{4} \).

Step 1: Identify the Type of Conic

The given polar equation is \( r = \frac{3}{4 - 4 \sin \theta} \). This is in the standard form of a conic section in polar coordinates: \( r = \frac{ed}{1 - e\sin\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \). Given the format, determine the type of conic by comparing with the standard forms.

Step 2: Determine the Eccentricity

From the equation \( r = \frac{3}{4 - 4 \sin \theta} \), we have \( ed = 3 \) and\( 1 - e\sin\theta = 4 - 4\sin\theta \). In the denominator, identify that \( 1 - e \sin \theta = 4 - 4\sin \theta \) implies \( e = 4 \). This indicates it is a hyperbola since \( e > 1 \).

Step 3: Calculate the Directrix

Since \( ed = 3 \) and \( e = 4 \), solve for the directrix \( d \): \( ed = 3 \) implies \( 4d = 3 \) so \( d = \frac{3}{4} \).

Step 4: Confirm the Conic and Its Components

The given equation has eccentricity \( e = 4 \) and a directrix \( d = \frac{3}{4} \), confirming it as a hyperbola. A hyperbola has its focus at the origin when it follows this polar form, which matches the given equation.

Key Concepts

Polar Coordinates

In the realm of mathematics, polar coordinates are a unique way of defining the position of a point in a plane. Unlike the Cartesian coordinate system, which uses x and y coordinates to plot points, polar coordinates use the distance from a reference point and an angle from a reference direction. Here's how it works:

• Imagine a point, called the origin, defined with polar coordinates as (0,0).
• The distance from this origin to any other point is denoted by \( r \), the radial coordinate.
• The angle measured from a reference line, usually the positive x-axis, to the line connecting the origin to the point is referred to as \( \theta \). This is the angular coordinate.

Conic sections in polar coordinates usually take the form \( r = \frac{ed}{1 - e\sin\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \), where allows capturing the unique characteristics of shapes like ellipses, hyperbolas, and parabolas based on their eccentricity \( e \). Understanding polar coordinates is crucial as it offers a simpler way to analyze complex curves, especially when dealing with rotational symmetry or radially varying properties.

Eccentricity

Eccentricity is a vital concept when studying conic sections. It describes how much a conic section deviates from being circular. This single value can categorize a conic into different types.

• An eccentricity \( e = 0 \) implies a perfect circle.
• When \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic becomes a parabola.
• When \( e > 1 \), the conic is identified as a hyperbola.

In the given equation, we determined the eccentricity as \( e = 4 \). Since \( e \) is greater than 1, this immediately identifies the conic as a hyperbola. The concept of eccentricity helps us understand the geometric properties and behavior of conic sections, thereby simplifying complex equations and enhancing problem-solving skills.

Directrix

Directrix is a line used in the definition of conic sections and helps to define their shape. For any point on the conic section, the ratio of its distance to the focus and to the directrix remains constant and is equal to the eccentricity. Here's how it applies to our problem:

• For the equation given, the directrix \( d \) is found using the relationship \( ed = 3 \).
• With an eccentricity \( e = 4 \), you solve for the directrix as \( d = \frac{3}{4} \).

The directrix helps in constructing and understanding conic sections more deeply. By knowing the directrix, one can derive various useful properties about the conic such as its orientation and key dimensions. Thus, it plays a pivotal role when dealing with conic sections, providing geometric insights beyond basic algebraic manipulation.

Hyperbola

A hyperbola is one of the four types of conic sections and is characterized by its open shape formed by two separate curves called branches. It emerges when slicing a double cone in a way that is steeper than the cone's side, resulting in two mirrored open curves. Here's a closer look:

• A hyperbola is a type of conic defined by the condition \( e > 1 \), where \( e \) is the eccentricity.
• Its standard form in polar coordinates is \( r = \frac{ed}{1 - e\sin\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \).
• In this form, \( ed = 3 \) and the eccentricity was identified as 4, confirming the hyperbola nature.

Hyperbolas have many interesting properties including asymptotes that define the angle at which the branches open and symmetry about the axes. Understanding hyperbolas is valuable in fields like astronomy and physics, where they help describe certain types of orbits and phenomena. Recognizing a hyperbola through its equation allows for deeper analysis and insights into its structural features.