Problem 6 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices. $$ r=\frac{3}{2-2 \sin \theta} $$

Step-by-Step Solution

Answer

Eccentricity is 2, indicating a hyperbola.

Step 1: Identify the General Form

The given equation is \( r=\frac{3}{2-2 \sin \theta} \). This can be compared to the general polar form of a conic section \( r = \frac{ed}{1 - e \sin \theta} \) where \( e \) is the eccentricity and \( d \) is the semi-latus rectum.

Step 2: Match and Find Eccentricity

Comparing \( r = \frac{3}{2-2 \sin \theta} \) to the general form, \( \frac{ed}{1 - e \sin \theta} \), we can identify that \( ed = 3 \), and \( -e = -2 \), meaning \( e = 2 \).

Step 3: Classify the Conic

The eccentricity \( e = 2 \) indicates that the conic is a hyperbola, as \( e > 1 \).

Step 4: Calculate the Semi-latus Rectum

Now that we have \( e \) and from \( ed = 3 \), we solve for \( d \): \( d = \frac{3}{e} = \frac{3}{2} \).

Step 5: Sketch the Graph and Label Vertices

When sketching the graph of the hyperbola in polar coordinates, note that the vertices are aligned with the direction of \( \theta \). For \( \sin \theta \), the angle \( \theta = \frac{\pi}{2} \) corresponds to one of the vertices. Substitute to find positions. The main vertices occur when \( 1 - 2\sin\theta \) changes sign, e.g., at \( \theta = \frac{\pi}{2} \) and at angles where the denominator equals zero.

Key Concepts

Eccentricity

Eccentricity is a crucial measure that helps us understand the shape of a conic section. It tells us how much the conic deviates from being a perfect circle. Given the formula of any conic in polar coordinates, eccentricity is represented by the variable "e".
- When the eccentricity, \( e \), is **equal to 1**, the conic is a **parabola**. - **Ellipse** when \( 0 < e < 1 \).- **Hyperbola** when \( e > 1 \). These rules determine the type of conic section based on its eccentricity value.
In our exercise, since \( e = 2 \), we classify the conic section as a **hyperbola**. This helps us understand that this conic will have two distinct branches, showing how much it differs from more circular shapes like ellipses.

Polar Coordinates

Polar coordinates offer a unique way of describing a point in the plane, especially useful for conic sections. Instead of using \((x, y)\) Cartesian coordinates, polar coordinates describe a point by its **distance** from the origin and the **angle** from the positive x-axis.
The formula for conic sections in polar coordinates is given by \( r = \frac{ed}{1 - e \sin \theta} \). Here:- \( r \) is the radial coordinate (distance from the origin).- \( \theta \) is the angular coordinate (angle with the x-axis).- **"e"**: Eccentricity of the conic.- **"d"**: Semi-latus rectum, representing the factor of the conic's width.
This system is particularly beneficial for analyzing circles, ellipses, parabolas, and hyperbolas. In our example, understanding the polar equation \( r=\frac{3}{2-2 \sin \theta} \) becomes easier as we translate between this radial angle-based system and traditional graphing.

Hyperbola

A hyperbola is a fascinating type of conic section that occurs when the eccentricity \( e \) is greater than 1. It features two separate "branches." The structure comes from slicing a double cone with a plane at a steep angle.
In terms of its polar equation, a hyperbola has the form \( r = \frac{ed}{1 - e \sin \theta} \). When graphed, each branch of the hyperbola moves further apart as \( r \) increases because of the influence of \( \theta \).
Hyperbolas have some remarkable properties, such as their asymptotes. These lines get infinitely close to the branches but never touch them. This property reflects the feature of hyperbolas possessing infinite extension in both directions.
In our exercise, drawing the hyperbola in polar coordinates helps visualize how quickly the hyperbola diverges from its central axes, characterized by an eccentricity of 2.

Vertices

Vertices in the context of a hyperbola are the closest points along each branch to the center. In polar coordinates, vertices lie along the axis of symmetry. When locating vertices from the polar equation, focus on values of \(\theta\) that make the denominator of the equation align with significant outputs for \( r \).
In \( r=\frac{3}{2-2 \sin \theta} \), vertices can be identified by substituting values into \(\theta\) that balance the terms effectively, such as \(\theta = \frac{\pi}{2}\). This angle, relative to the polar axis, shows where the graph reaches its "tip".
Each vertex gives a distinct insight into the curve's behavior, representing the hyperbola's most stretched-out points. When sketching, these are crucial reference points in identifying the conic's layout and orientation.