Problem 51 - College Algebra

Question

Find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=4 ; e=\frac{3}{2}\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{6}{1 - \frac{3}{2}\sin(\theta)} \).

Step 1: Understanding Conic Sections and Polar Equations

Conic sections are the curves obtained by intersecting a double cone with a plane. For polar equations of conics, with focus at the origin, the formula is given by \( r = \frac{ed}{1 - e\sin(\theta)} \) for vertical directrix. Here, 'e' is the eccentricity and 'd' is the distance from the focus to the directrix.

Step 2: Identifying Type of Conic

The given eccentricity is \( e = \frac{3}{2} = 1.5 \), which is greater than 1. This indicates that the conic is a hyperbola, since hyperbolas have an eccentricity greater than 1. Additionally, the directrix \( y = 4 \) is vertical.

Step 3: Establishing Parameters

The directrix is given as \( y = 4 \). The distance 'd' from the focus (origin) to the directrix in the polar equation is \( d = 4 \). This is critical to formulating the equation.

Step 4: Writing the Polar Equation

The polar equation for a conic with a vertical directrix \( y = 4 \) and eccentricity \( e = \frac{3}{2} \) is defined as \( r = \frac{ed}{1 - e\sin(\theta)} \). By substituting \( e = \frac{3}{2} \) and \( d = 4 \), we obtain: \( r = \frac{\frac{3}{2} \times 4}{1 - \frac{3}{2}\sin(\theta)} = \frac{6}{1 - \frac{3}{2}\sin(\theta)} \).

Key Concepts

Conic Sections

Conic sections are fascinating curves that arise from the intersection of a plane and a double-napped cone. Depending on the angle at which the plane intersects the cone, we can obtain different shapes: circles, ellipses, parabolas, or hyperbolas. These shapes are collectively known as conics and have various properties and applications, especially in the field of geometry.
Conic sections are classified based on the eccentricity value, which distinguishes between the different curves:

• Circle: An eccentricity of 0.
• Ellipse: An eccentricity between 0 and 1.
• Parabola: An eccentricity of exactly 1.
• Hyperbola: An eccentricity greater than 1.

Understanding conic sections can help us in solving mathematical problems related to curves and analyzing their behaviors in different contexts.

Eccentricity

Eccentricity is a fundamental concept when discussing conic sections. It measures how much a conic section deviates from being circular. In other words, it's a number that helps us identify the type of conic section we are working with.
The value of eccentricity, denoted as 'e', determines the shape:

• If the eccentricity, e, is below 1, the conic is more like a squashed circle (ellipse).
• If e equals 1, the conic is a parabola.
• If e is greater than 1, the conic takes the form of a hyperbola, characterized by its two separate branches.

For the given exercise, an eccentricity of 1.5 tells us we are dealing with a hyperbola, as it exceeds 1, indicating the conic opens outward and has multiple separated components.

Directrix

The directrix is a straight line associated with a conic section, providing a reference for defining the curve. It is one of the components used to define conics such as parabolas, ellipses, and hyperbolas, alongside their focus and eccentricity.
For hyperbolas and ellipses, the directrix helps in developing an equation in polar form. In the polar coordinate system, the directrix becomes crucial, as it is paired with the concept of focus. The polar equation of a conic with a focus at the origin can be established using the directrix and eccentricity.
In our context, the directrix is given as a vertical line, represented by the equation 'y = 4'. This means it's a horizontal line four units away from the origin in the Cartesian plane. It helps guide us in forming the equation of the polar graph of the hyperbola.

Hyperbola

Hyperbolas are a type of conic section that appear when the plane cuts through both nappes of the cone. They are defined by having two distinct branches, opening outward in opposite directions.
A hyperbola is characterized by an eccentricity greater than 1, which means it's more elongated than a circle or ellipse. For polar equations, hyperbolas follow the formula: \[ r = \frac{ed}{1 - e \sin(\theta)} \]where 'e' is the eccentricity and 'd' is the distance from the focus to the directrix.
In a hyperbola, the distance between the foci is greater than the distance between the vertices, giving it a distinct shape. Understanding hyperbolas is essential for tackling problems related to advanced geometry, orbit paths of celestial bodies, and various physics simulations.