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Magazine Chapter 10: Problem 20
(a) Graph the conics \(r=e d /(1+e \sin \theta)\) for \(e=1\) and various values of \(d .\) How does the value of \(d\) affect the shape of the conic? (b) Graph these conics for \(d=1\) and various values of \(e\). How does the value of \(e\) affect the shape of the conic?
Short Answer
Expert verified
The value of \( d \) changes the size/position; \( e \) determines the conic type.
Step by step solution
01
Understanding the Conic Equation
The conic equation given is \( r = \frac{ed}{1 + e \sin \theta} \). Here, \( e \) is the eccentricity, and \( d \) is a constant related to directrix distance. The shape of the conic section (circle, parabola, ellipse, or hyperbola) depends on the value of \( e \).
02
Analyzing the Effect of 'd' with Fixed 'e=1'
For \( e = 1 \), the conic is initially a parabola since the eccentricity \( e = 1 \) for parabolas. Changing \( d \) does not alter the type of the conic (it remains a parabola), but affects its size and position because \( d \) represents a scaling factor related to the directrix.
03
Graphing for Various 'd' Values
Plot the equation \( r = \frac{d}{1 + \sin \theta} \) for several values of \( d \) (e.g., 0.5, 1, 2) while keeping \( e = 1 \). Observe the size and scale of the parabola for different \( d \). The conic's opening widens with decreasing \( d \) and tightens with increasing \( d \).
04
Analyzing the Effect of 'e' with Fixed 'd=1'
For \( d = 1 \), the shape changes with \( e \):- If \( e < 1 \), the conic is an ellipse.- If \( e = 1 \), the conic is a parabola.- If \( e > 1 \), the conic is a hyperbola.
05
Graphing for Various 'e' Values
Plot the equation \( r = \frac{e}{1 + e \sin \theta} \) for values of \( e \) (e.g., 0.5, 1, 2). Observe that:- Ellipses occur for \( e < 1 \).- A parabola forms for \( e = 1 \).- Hyperbolas arise for \( e > 1 \).The 'eccentricity' \( e \) controls the type of conic, altering its fundamental nature.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Eccentricity
Eccentricity is a measure that defines the shape of a conic section. It is denoted by \( e \) and dictates whether the conic is an ellipse, a parabola, or a hyperbola.
- When \( e = 1 \), the conic is a parabola. This is a special condition where the conic curves exactly such that it is equidistant at all points from a fixed point called the focus and a fixed line known as the directrix.
- If \( e < 1 \), it is an ellipse. An ellipse is a stretched circle, where the eccentricity measures how much it deviates from being circular. The closer \( e \) is to 0, the more circular the ellipse appears.
- For \( e > 1 \), the conic is a hyperbola. This means the shape splits into two symmetrical curves facing away from each other around the foci.
Polar Coordinates
Polar coordinates are a system where each point on a plane is determined by an angle and a distance from a reference point, usually called the origin or pole. This system is especially useful for graphing conic sections, particularly because it simplifies the equations used to describe them.
- The reference point (analogous to the origin in Cartesian coordinates) is known as the pole.
- The distance from the pole to a point is denoted by \( r \), and the angle from the positive x-axis to the line segment from the pole to the point is denoted by \( \theta \).
Parabola
A parabola is a conic section that can be understood as a path or the locus of points equidistant from a fixed point (focus) and a fixed straight line (directrix). For parabolas, the eccentricity \( e \) is always equal to 1, and this unique property defines it from other conic sections.
- When graphed in polar coordinates with \( e = 1 \), the equation simplifies to \( r = \frac{d}{1 + \sin \theta} \), showcasing the parabolic shape.
- Adjusting the constant \( d \) affects the parabolas' size and positioning but not its fundamental parabolic nature.
Ellipse
An ellipse is another type of conic section, where \( e < 1 \). It's like a squished or stretched circle and is defined as the locus of all points for which the sum of the distances to two fixed points (the foci) is constant.
- The equation for an ellipse in polar coordinates when \( e < 1 \) remains similar but tweaks its parameters to obtain the specific elliptical form.
- The closer the eccentricity is to zero, the more the ellipse resembles a perfect circle.
Hyperbola
A hyperbola emerges as a conic section when the eccentricity \( e > 1 \). This means it forms two separate branches that mirror each other across a center axis.
- The polar equation changes to reflect the hyperbolic nature as \( e \) surpasses 1, creating this open-ended curve.
- In this shape, the difference, rather than the sum, of distances from any point on the hyperbola to two foci is constant.
