Problem 54 - College Algebra

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=2 ; e=2.5\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{5}{1 - 2.5 \cdot \sin(\theta)} \).

Step 1: Understand Conics in Polar Form

The polar equation of a conic section with the focus at the origin is given by the general formula: \[ r = \frac{ed}{1 - e \cdot \sin(\theta)} \] where \( e \) is the eccentricity and \( d \) is the distance from the origin to the directrix. The type of conic is determined by the eccentricity: \( e=1 \) is a parabola, \( e<1 \) is an ellipse, and \( e>1 \) is a hyperbola. Here, \( e = 2.5 \).

Step 2: Determine the Distance to Directrix

The given directrix is \( y = 2 \). Since the directrix is a horizontal line, the distance from the focus at the origin to this directrix is \( |2| = 2 \). Thus, the value of \( d \) is 2.

Step 3: Substitute Values into the Polar Equation

Substitute \( e = 2.5 \) and \( d = 2 \) into the polar form of the conic equation: \[ r = \frac{2.5 \times 2}{1 - 2.5 \cdot \sin(\theta)} \] Simplifying, we get: \[ r = \frac{5}{1 - 2.5 \cdot \sin(\theta)} \].

Step 4: Verify the Equation Type

Given that the eccentricity \( e = 2.5 \) is greater than 1, the conic is a hyperbola. This aligns with our derived equation indicating an expansion or opening in opposite directions.

Key Concepts

Understanding Eccentricity in Conics

Eccentricity, often denoted by the letter \( e \), is a crucial concept when it comes to understanding conic sections. It essentially measures how much a conic section deviates from being a perfect circle. Here's a simple breakdown:

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

In the provided exercise, the eccentricity \( e \) is 2.5, which is greater than 1. This indicates that the conic section is a hyperbola. In very basic terms, a hyperbola consists of two separate branches, each appearing as a mirror image of the other about an axis.

Clarifying the Directrix of a Conic

The directrix of a conic section plays an integral role in defining its shape along with the eccentricity. It is a fixed line used in the description of a conic section. For conics described in polar form like in this exercise, the directrix helps determine the geometry of the conic.

The distance to the directrix, denoted as \( d \), is measured perpendicular to the line from the focus. In our exercise, the directrix is given as \( y = 2 \). This means it is a horizontal line located two units vertically from the focus at the origin. Therefore, the distance \( d \) is simply 2.

The directrix and the value of \( d \) are used to set up the equation for the conic, solidifying the conic's position in the polar coordinate system.

Exploring Types of Conic Sections

Conic sections are the curves obtained by intersecting a plane with a cone. These sections are divided into several types, each corresponding to certain geometric conditions. Let's explore these different types:

• Ellipse: A conic section where the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. Characterized by eccentricity \( 0 < e < 1 \).
• Parabola: A conic where each point is equidistant from a single focus and the directrix. This is the case when \( e = 1 \).
• Hyperbola: Formed when the difference of the distances from any point on the hyperbola to two fixed points (foci) is constant. It occurs when \( e > 1 \), making their shape distinguishable by two separate arms or branches.

In our specific exercise, the given eccentricity was \( e = 2.5 \), so the resulting conic is a hyperbola. Each type of conic section has unique properties and finds various applications in science and engineering, such as satellite dish designs (parabolas) or planetary orbits (ellipses).