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Magazine Chapter 11: Problem 38
Include the directrix that corresponds to the focus at the origin. Label the vertices with appropriate polar coordinates. Label the centers of the ellipses as well. $$r=\frac{6}{2+\cos \theta}$$
Short Answer
Expert verified
The directrix is \( x = -6 \) and the vertex is at \( (6, \pi) \).
Step by step solution
01
Identifying the Conic Section
The given polar equation is \( r = \frac{6}{2 + \cos \theta} \). This format typically represents a conic section in polar coordinates: \( r = \frac{ed}{1 + e\cos \theta} \) or \( r = \frac{ed}{1 + e\sin \theta} \). Since it uses \( \cos \theta \), the conic opens along the polar axis, suggesting a horizontal directrix parallel to the polar axis.
02
Determining the Type of Conic Section
For the equation \( r = \frac{ed}{1 + e\cos \theta} \), the eccentricity \( e \) can be found by reformatting it into the standard form. Here, \( ed = 6 \) and the denominator \( 2 + \cos \theta \) means \( 1 + e\cos \theta = 2 + \cos \theta \). Thus, \( e = 1 \), which confirms the conic is a parabola as \( e = 1 \) signifies a parabola.
03
Finding Directrix and Focus
The directrix of a parabola where the eccentricity \( e = 1 \) and the focus is at the origin, is given by its distance from the focus, \( d = \frac{6}{e} = 6 \). Therefore, the directrix is the line \( x = -6 \), determined by rearranging \( r = \frac{6}{2 + \cos \theta} \) into cartesian form.
04
Labeling the Vertex
In polar coordinates, the vertex of the parabola is at the point where \( r \) is minimized while \( \cos \theta = -1 \) (since that maximizes the denominator). Substituting \( \theta = \pi \) (where \( \cos \theta = -1 \)) into the equation gives \( r = \frac{6}{2 - 1} = 6 \). Thus, the polar coordinate for the vertex is \((6, \pi)\).
05
Labeling Centers
Since this conic is a parabola and not an ellipse, it does not have a center. The parabola's vertex \((6, \pi)\) serves as the key point of reference, indicative of its position and direction.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Conic Sections
Conic sections are curves obtained by intersecting a cone with a plane. These curves include ellipses, parabolas, hyperbolas, and circles. Each conic section has a unique set of features that differentiate it from others.
- An **ellipse** is shaped like an elongated circle. It includes circles as a special case.
- A **parabola** is a U-shaped curve that mirrors symmetrically about its axis.
- A **hyperbola** consists of two separate curves that are mirror images of each other.
- A **circle** is a special ellipse where the two foci are at the same point.
Exploring the Parabola
A parabola is a specific type of conic section characterized by its "U" shaped curve. When a conic section's eccentricity is equal to 1, it becomes a parabola.
This shape has a single point, known as the vertex, and a line called the axis of symmetry that divides it into two equal parts. The standard form in polar coordinates for a parabola looks like this:
This shape has a single point, known as the vertex, and a line called the axis of symmetry that divides it into two equal parts. The standard form in polar coordinates for a parabola looks like this:
- Focus at the origin
- Equation: \( r = \frac{ed}{1 + e\cos \theta} \)
The Role of Eccentricity
Eccentricity, often denoted as **e**, describes how much a conic section deviates from being circular. It is a non-negative real number that helps determine the shape of the conic section. Different conic sections have different ranges of eccentricity values:
- **e = 0** for a circle
- **0 < e < 1** for an ellipse
- **e = 1** for a parabola
- **e > 1** for a hyperbola
Defining the Directrix
The directrix of a conic section serves as a reference line from which distances are measured. For a parabola, the distance from any point on the parabola to the directrix is proportional to its distance from the focus.
In our exercise, the directrix is a line that helps define the position and orientation of our parabola. Specifically, with the given polar equation \( r = \frac{6}{2 + \cos \theta} \), the directrix is found at **x = -6**.
In our exercise, the directrix is a line that helps define the position and orientation of our parabola. Specifically, with the given polar equation \( r = \frac{6}{2 + \cos \theta} \), the directrix is found at **x = -6**.
- This means the curve is aligned horizontally in the direction.
- The directrix being located on the left side of the origin helps orient the parabola in a manner that reflects its geometric properties with respect to the focus.
