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Magazine Chapter 12: Problem 33
(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$ r=\frac{6}{2+\sin \theta} $$
Short Answer
Expert verified
The eccentricity is \(\frac{1}{2}\), indicating the conic is an ellipse.
Step by step solution
01
Identify the Conic Equation Type
We are given the polar equation \( r = \frac{6}{2 + \sin \theta} \). Compare this with the standard form \( r = \frac{l}{1 + e \sin \theta} \). The given equation can be rewritten in this form: \( r = \frac{6}{2 + \sin \theta} = \frac{6/2}{1 + (1/2) \sin \theta} \). Thus, \( l = 3 \) and \( e = \frac{1}{2} \).
02
Calculate and Interpret the Eccentricity
The eccentricity \( e = \frac{1}{2} \) is important in identifying the type of conic. If \( e < 1 \), the conic is an ellipse; if \( e = 1 \), it is a parabola; and if \( e > 1 \), it is a hyperbola. Here \( e = \frac{1}{2} < 1 \), so the conic is an ellipse.
03
Express the Conic in Cartesian Coordinates
For a deeper understanding, express the given conic in Cartesian form. The polar equation \( r = \frac{6}{2 + \sin \theta} \) can be converted by multiplying both sides by \( 2 + \sin \theta \), leading to \( r(2 + \sin \theta) = 6 \). Substitute \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{\sqrt{x^2 + y^2}} \) to find the Cartesian form. The algebraic steps are complex and may be unnecessary since the type is known from \( e \).
04
Sketch the Conic and Label the Vertices
Focus on sketching the ellipse. In polar coordinates, for the maximum \( \theta = \frac{\pi}{2} \), \( r = \frac{6}{2 + 1} = 2 \). For \( \theta = -\frac{\pi}{2} \), \( r = \frac{6}{2 - 1} = 6 \). These provide two key points, which should be mirrored across the center to fully depict the ellipse's major axis centered at the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eccentricity
Eccentricity is a fundamental concept when discussing conics, as it helps us understand a conic section's shape. It is a parameter that defines the extent to which a conic section deviates from being circular. The eccentricity (\(e\)) can have different values, each indicating a specific type of conic:
- For an ellipse, \(0 < e < 1\).
- For a parabola, \(e = 1\).
- For a hyperbola, \(e > 1\).
- For a circle, a special case of an ellipse, \(e = 0\).
Polar Coordinates
Polar coordinates are an alternative to Cartesian coordinates, and they are particularly useful for plotting curves like conics. In this system, each point is determined by a distance from a reference point (the pole) and an angle from a reference direction, usually the positive x-axis.
- The coordinate pair \((r, \theta)\) gives the position of a point, where \(r\) is the radius or distance from the pole, and \(\theta\) is the angle with respect to the reference direction.
- This system is advantageous for systems where symmetry about a point is present, such as circles and ellipses.
Ellipse
An ellipse is one of the principal conic sections and is characterized by its elliptical shape. It may be visualized as a squashed circle, where all points satisfy the equation \(r = \frac{l}{1 + e \sin \theta}\) or its equivalent Cartesian form. Key properties of an ellipse include:
- Two focal points (foci).
- The total distance from any point on the ellipse to the two foci is constant.
- For angles \(\theta = \frac{\pi}{2}\) and \(\theta = -\frac{\pi}{2}\), the ellipse's radius is minimized and maximized, respectively.
- Vertices are the endpoints of the major axis, which in polar form indicates the maximum and minimum radii.
Cartesian Coordinates
Cartesian coordinates, also known as rectangular coordinates, describe a point's position in terms of a \((x, y)\) pair. The system is straightforward and familiar, used to plot graphs on a two-dimensional plane.
- The x-axis and y-axis define the plane, dividing it into four quadrants.
- The transformation of polar coordinates into Cartesian coordinates involves using relationships \(x = r \cos \theta\) and \(y = r \sin \theta\).
- This conversion is often necessary for representing equations or functions initially described in polar form for easier graphing through familiar methods.
