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Chapter 13: Problem 77

Show that for eccentricity equal to one in Equation 13.10 for conic sections, the path is a parabola. Do this by substituting Cartesian coordinates, \(x\) and \(y,\) for the polar coordinates, \(r\) and \(\theta,\) and showing that it has the general form for a parabola, \(x=a y^{2}+b y+c\).

Short Answer

Expert verified
When the eccentricity, \(e\), is equal to 1 in Equation 13.10 for conic sections, the equation in polar coordinates is \(r = \frac{p}{1 + \cos(\theta)}\). Converting this to Cartesian coordinates and simplifying, we obtain the equation \(x = p - \frac{1}{4p} y^2\), which has the general form of a parabola, \(x = a y^2 + b y + c\), with \(a = -\frac{1}{4p}\), \(b = 0\), and \(c = p\). Therefore, the path is a parabola when the eccentricity is equal to one.

Step by step solution

01

Write the equation for conic sections in polar coordinates

The general equation for conic sections in polar coordinates is given by: \(r = \frac{p}{1 + e \cdot \cos(\theta)}\) Here, \(r\) is the radial distance, \(p\) is the semi-latus rectum, \(e\) is the eccentricity, and \(\theta\) is the polar angle.
02

Substitute the value of eccentricity

We are given that the eccentricity, \(e\), is equal to 1. Substituting the value of \(e = 1\) in the equation, we get: \(r = \frac{p}{1 + \cos(\theta)}\)
03

Convert the equation from polar to Cartesian coordinates

To convert the equation from polar to Cartesian coordinates, we use the following conversions: 1. \(x = r \cos(\theta)\) 2. \(y = r \sin(\theta)\) Now, express \(r\) in terms of \(x\) and \(y\) by substituting the above conversions: \(r = \frac{p}{1 + \cos(\theta)} \Rightarrow x = \frac{p \cos(\theta)}{1 + \cos(\theta)}\) Next, express \(\cos(\theta)\) in terms of \(x\) and \(y\): \(\cos(\theta) = \frac{x}{r} \Rightarrow x = r \cos(\theta)\) Substituting the given value of \(r\) from the above equation: \(x = \frac{p \cos(\theta)}{1 + \cos(\theta)} \cdot \cos(\theta)\) Now, express \(\sin(\theta)\) in terms of \(x\) and \(y\): \(\sin(\theta) = \frac{y}{r} \Rightarrow y = r \sin(\theta)\) Again, substituting the given value of \(r\) from the above equation: \(y = \frac{p \cos(\theta)}{1 + \cos(\theta)} \cdot \sin(\theta)\)
04

Compare the obtained equation to the general form of a parabola

Now, we need to show that the obtained equation has the general form for a parabola, which is: \(x = a y^2 + b y + c\) Divide the equation of \(y\) from Step 3 by the equation of \(x\) from Step 3: \(\frac{y}{x} = \frac{\sin(\theta)}{\cos(\theta)}\) \(\frac{y}{x} = \tan(\theta)\) Express \(\tan(\theta)\) in terms of \(x\) and \(y\): \(\tan(\theta) = \frac{y/x}{1 - (y/x)^2}\) Now, substitute the expression of \(\tan(\theta)\) in the equation of \(y\) from Step 3: \(y = \frac{p \cos(\theta)}{1 + \cos(\theta)} \cdot \sin(\theta)\) Using the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), we get: \(y^2 = p^2 + 2px - x^2\) Thus, the obtained equation in Cartesian coordinates is: \(x = p - \frac{1}{4p} y^2\) Comparing this equation to the general form of a parabola, we see that it has the form \(x = a y^2 + b y + c\) with \(a = -\frac{1}{4p}\), \(b = 0\), and \(c = p\). Therefore, the path is a parabola when the eccentricity is equal to one.

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Key Concepts

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

Eccentricity
Eccentricity is a parameter that determines the shape of a conic section. It's denoted by the letter \( e \) and influences whether the conic section is a circle, ellipse, parabola, or hyperbola. When \( e = 1 \), we have a parabola. This is a unique property of parabolas because it represents the boundary between ellipses \((e < 1)\) and hyperbolas \((e > 1)\).

The eccentricity tells us how "stretched" a conic section is. Understanding the value of \( e \) helps identify the conic section without needing to plot it. Here鈥檚 a simple breakdown:
  • If \( e = 0 \), the conic is a circle.
  • If \( 0 < e < 1 \), the conic is an ellipse.
  • If \( e = 1 \), the conic is a parabola.
  • If \( e > 1 \), the conic is a hyperbola.
Each conic has unique properties useful in different applications, like satellite dish designs focusing on parabolas.
Polar Coordinates
Polar coordinates provide a way to represent points in a plane by their distance from a reference point, called the pole, and the angle from a reference direction, usually the positive x-axis. This can be particularly useful for describing curves and conic sections, like ellipses and parabolas.

The two components of polar coordinates are:
  • \( r \), the radial distance from the pole.
  • \( \theta \), the angular direction from a reference line.
When dealing with conic sections, polar coordinates are beneficial as they simplify the mathematical expressions, like \( r = \frac{p}{1 + e \cdot \cos(\theta)} \). For a parabola (\( e = 1 \)), this simplifies to \( r = \frac{p}{1 + \cos(\theta)} \). Understanding these conversions is key when shifting between coordinate systems.
Parabola
A parabola is one of the conic sections formed when a cone is intersected by a plane parallel to its side. It has a distinct U-shaped figure. The general equation defining a parabola in Cartesian coordinates is \( x = a y^2 + b y + c \).

The parabolic shape has a special property: any ray parallel to its axis of symmetry that reflects off its surface will pass through a unique point called the focus. This property is utilized in parabolic antennas and telescopes.

When the eccentricity \( e = 1 \), the polar equation of a parabola \( r = \frac{p}{1 + \cos(\theta)} \), can be converted to its Cartesian form, \( x = a y^2 + b y + c \), showing its unique characteristics in different coordinates. Parabolas have applications ranging from natural phenomena to engineering, making the comprehension of their properties crucial.
Cartesian Coordinates
Cartesian coordinates are the most conventional method of expressing the position of points in space. This coordinate system uses two perpendicular axes, typically labeled the x and y-axes in a plane, to determine the position of a point.

These coordinates give us a straightforward representation of geometric figures, enhancing the understanding of algebraic solutions and graphing. For the equation of a parabola, \( x = a y^2 + b y + c \), the utilization of Cartesian coordinates allows for easier plotting and solving of equations.

When converting from polar to Cartesian coordinates, the transformations used are:
  • \( x = r \cos(\theta) \)
  • \( y = r \sin(\theta) \)
These conversions facilitate the interpretation and manipulation of the equations of conic sections, like transforming \( r = \frac{p}{1 + \cos(\theta)} \) into its parabolic form in Cartesian coordinates, thereby connecting different mathematical representations of the same geometric shapes.

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