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Chapter 9: Problem 11

Identify the conic section given by each of the equations. $$r=\frac{3}{1-0.6 \sin \theta}$$

Short Answer

Expert verified
The conic section described by the equation \(r=\frac{3}{1-0.6 \sin \theta}\) is a hyperbola.

Step by step solution

01

Identify the equation

Looking at the given equation \(r=\frac{3}{1-0.6 \sin \theta}\), one can see that it fits the standard form for a conic section in polar coordinates \(r = \frac{e}{1 \pm e \sin (\theta -\phi)}\) where there is a sine function in the equation, and \(\phi = 0\).
02

Determine the eccentricity

The eccentricity is given by the value of e in the equation. Here, e = 3. There is a multiplying factor of 0.6 before the sine function, this gives the eccentricity of the conic as \(3/0.6 = 5\).
03

Identify the conic section

With the computed eccentricity of 5, which is greater than 1, the conic section is a hyperbola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This is different from the Cartesian coordinate system that uses two values that represent a point's location on the x and y axes.

In polar coordinates, you have:
  • A radial coordinate, denoted as \( r \), which represents the distance from the origin (or pole) to the point.
  • An angular coordinate, \( \theta \), measured in radians or degrees, that indicates the direction of the point from the positive x-axis.
This system is particularly useful when dealing with problems with circular or rotational symmetry, like many scenarios in physics. When identifying conic sections, equations can be rewritten in polar form to highlight properties such as symmetry and eccentricity, which we'll discuss next.
Eccentricity
Eccentricity is an essential concept in understanding conic sections. It is a measure that describes how much a conic section deviates from being circular.

A conic section can be characterized by its eccentricity (\( e \)), which determines its shape:
  • If \( e = 0 \), the conic is a circle.
  • If \( 0 < e < 1 \), it's an ellipse.
  • If \( e = 1 \), the conic is a parabola.
  • If \( e > 1 \), you have a hyperbola.
In our original problem, we calculate eccentricity by rearranging the given equation to identify \( e \). With an eccentricity of 5, we can conclude that the conic section is a hyperbola, showing significant deviation from a circle.
Hyperbola
A hyperbola is one of the four types of conic sections, derived when intersecting a plane with a double cone. It consists of two disconnected curves, called branches, which mirror each other.

Characteristics of a hyperbola include:
  • Open curves that extend infinitely in both directions.
  • Asymptotes are straight lines that the hyperbola approaches but never touches or crosses.
  • An eccentricity greater than 1, as seen in our previous calculation with \( e = 5 \).
Hyperbolas have many applications such as in navigation systems and understanding the orbits of celestial bodies. They can be expressed in different forms including polar coordinates, which is useful in describing situations involving angles and magnitudes.

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