Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 18
15鈥�22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{10}{3-2 \sin \theta}$$
Short Answer
Expert verified
The conic is a hyperbola with eccentricity 2.
Step by step solution
01
Identify the form of the conic equation
The given polar equation is \( r = \frac{10}{3 - 2 \sin \theta} \). This equation compares to the general form of a conic in polar coordinates, \( r = \frac{ed}{1 - e \sin \theta} \), which indicates it's a conic with respect to a focus at the pole.
02
Determine the parameters 'e' and 'd'
For comparison, the given equation can be rewritten to match \( r = \frac{ed}{1 - e \sin \theta} \). Here, \( ed = 10 \) and \( 1 - e \sin \theta = 3 - 2 \sin \theta \). From matching coefficients, we have:1. \( ed = 10 \)2. \( e = \frac{2}{1} = 2 \)
03
Identify the eccentricity
From the step above, we've determined that the eccentricity \( e = 2 \). The value of eccentricity reveals the type of conic. For conics:- If \( e = 1 \), it's a parabola.- If \( e < 1 \), it's an ellipse.- If \( e = 0 \), it's a circle.- If \( e > 1 \), it's a hyperbola.Since \( e = 2 \) which is greater than 1, the conic is a hyperbola.
04
Identify the conic
The conic is identified as a hyperbola since the eccentricity \( e = 2 \) and is greater than 1.
05
Sketch the conic
To sketch the hyperbola, we note that the directrix is vertical since the equation is in the form \( r = \frac{ed}{1 - e \sin \theta} \). The vertices are found from polar coordinates by considering the extreme values of \( r \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eccentricity
In the realm of conic sections, eccentricity is a fundamental property that helps in identifying and distinguishing various conic shapes. It is denoted by the symbol \( e \). The eccentricity determines the shape of the conic section:
- If \( e = 0 \), the conic is a circle, which is the most "balanced" shape.
- If \( 0 < e < 1 \), the shape is an ellipse, which is somewhat stretched.
- If \( e = 1 \), it indicates a parabola, showing a specific open curve.
- If \( e > 1 \), the conic section is a hyperbola, characterized by two separate branches.
Conic Sections
Conic sections are the curves obtained by slicing a double cone. These sections capture a variety of shapes:
- A circle, which is a curve where all points are equidistant from the center.
- An ellipse, which resembles a "squashed" circle, with two focal points.
- A parabola, which has a distinctive curved shape with one focal point.
- A hyperbola, consisting of two mirrored, open curves known as branches.
Hyperbola
A hyperbola is a fascinating type of conic section that has unique properties compared to other conic sections. It consists of two disconnected curves called branches. These branches are symmetrical around a center point.
- The eccentricity \( e \) of a hyperbola is always greater than 1.
- In the context of the given polar equation \( r = \frac{10}{3 - 2 \sin \theta} \), the equation of a hyperbola indicates that as \( \theta \) changes, \( r \) changes, allowing the branches to extend indefinitely.
- Hyperbolas can often model systems involving orbits and escape velocities, making them crucial in celestial mechanics.
Polar Equation
Polar equations are a beautiful way to represent conics using the polar coordinate system. They relate a radial distance \( r \) and an angular coordinate \( \theta \).
- The general form for a conic in polar coordinates with a focus at the pole is \( r = \frac{ed}{1 - e \sin \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \).
- This polar form is particularly useful for identifying and describing conic sections when a focus is considered one of the poles.
- In our equation \( r = \frac{10}{3 - 2 \sin \theta} \), we derived \( e = 2 \) and thus identified it as a hyperbola. The structure provides insight into how \( r \) changes as \( \theta \) rotates.
Vertex of Conic
The vertices of a conic are crucial points that help in sketching and understanding the overall shape of the conic section. In the case of a hyperbola, each branch has its own vertex.
- A vertex is the point on each branch where the conic section comes closest to or is furthest from its center.
- From the polar equation \( r = \frac{10}{3 - 2 \sin \theta} \), vertices can be identified by examining the extreme values of \( r \), which inform how far the curves extend.
- In practical terms, the vertices are essential for providing a framework to accurately sketch the shape of the hyperbola and understand its symmetry.
