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Chapter 11: Problem 12

Identify the conic defined by each polar equation. Also give the position of the directrix. \(r=\frac{6}{8+2 \sin \theta}\)

Short Answer

Expert verified
The conic is a hyperbola with the directrix at \( d = 3 \).

Step by step solution

01

- Identify the standard form of the polar equation

Recall that the standard form of a polar equation of a conic section is given by: \( r = \frac{ed}{1 + e \sin \theta} \) or \( r = \frac{ed}{1 + e \cos \theta} \).
02

- Compare given equation to standard form

Compare the given equation \( r = \frac{6}{8+2 \sin \theta} \) to the standard form \( r = \frac{ed}{1 + e \sin \theta} \). Notice that it can be rewritten as \( r = \frac{6}{8 + 2 \sin \theta} \).
03

- Extract the values of e and d

The standard form of the equation \( r = \frac{ed}{1 + e \sin \theta} \) suggests that: \(ed = 6\) and \(1 + e \sin \theta = 8 + 2 \sin \theta\).
04

- Solve for eccentricity e

To find e, recognize that the term involving \(\sin \theta\) in the denominator fits the standard form when comparing coefficients: \(1 + e \sin \theta = 8 + 2 \sin \theta\) gives \( e = 2 \).
05

- Solve for the directrix d

Using \(e = 2\), substitute back into the equation \( ed = 6 \):\( 2d = 6 \Rightarrow d = 3\).
06

- Identify the conic and the position of the directrix

Since the eccentricity \(e = 2\), which is greater than 1, the conic described is a hyperbola. The directrix is located at \( d = 3 \).

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Key Concepts

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

Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Each type of conic section has unique characteristics depending on the angle of the plane with respect to the cone. The four main types of conic sections are:

  • Circle
  • Ellipse
  • Parabola
  • Hyperbola
In polar coordinates, conic sections can be described by equations of the form:\r \(r = \frac{ed}{1 + e \, \sin \, \theta}\) or \(r = \frac{ed}{1 + e \, \cos \, \theta}\).

Here, e (eccentricity) and d (directrix) are parameters that characterize the conic.
Eccentricity
Eccentricity, denoted by e, is a non-negative real number that describes how much a conic section deviates from being circular.

Depending on the value of e:

  • If e = 0, the conic is a circle.
  • If 0 < e < 1, it is an ellipse.
  • If e = 1, it is a parabola.
  • If e > 1, it is a hyperbola.
In the exercise, the given polar equation \(r = \frac{6}{8 + 2 \, \sin \, \theta}\) can be rewritten to fit the standard polar form. Comparing it and solving for e shows us that e (eccentricity) = 2, which confirms the given conic is a hyperbola.
Directrix
The directrix of a conic section is a fixed line used in describing and defining the conic. In the context of polar coordinates, the directrix helps determine the shape and position of the conic section.

For the equation
  • \(r = \frac{ed}{1 + e \, \sin \, \theta}\)
the directrix is related to the constant d. Higher values of d affect the width of the hyperbola.

In our exercise, by comparing and simplifying the given polar equation to standard form, we identified that d (directrix) = 3, placing the directrix three units away from the origin.
Hyperbola
A hyperbola is a type of conic section that appears when the plane intersects both nappes of the cone. Hyperbolas have two branches, which are mirror images of each other.

Key features of a hyperbola include:
  • Vertices: points where each branch is closest to the other.
  • Foci: two fixed points used to define the hyperbola.
  • Asymptotes: lines that the branches approach but never touch.
In polar form, a hyperbola is identified when the eccentricity e > 1.

In the given equation, since the eccentricity e (eccentricity) = 2 (which is greater than 1), the conic section is indeed a hyperbola. Its branches symmetrically arrange around the directrix located at d = 3.

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