Explanation Given: r = d e 1 − e cos θ , r = − d e 1 + e cos θ and f 1 ( θ ) = − f 2 ( θ + π ) , r = f 1 ( θ ) and r = f 2 ( θ ) . Calculation: It is given that f 1 ( θ ) = − f 2 ( θ + π ) so, evaluate f 2 ( θ + π ) by substituting θ = θ + π and we get: f 2 ( θ + π ) = − d e 1 + e cos ( θ + π ) = − d e 1 + e ( − cos θ ) ( cos ( θ + π ) = − cos θ ) = − d e 1 − e cos θ = − f 1 ( θ ) Thus, it is shown that r = f 1 ( θ ) and r = f 2 ( θ ) define the same curves in polar coordinates if f 1 ( θ ) = − f 2 ( θ + π ) ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 12, Problem 40CRE

To determine

That $r = f_{1} (θ)$ and $r = f_{2} (θ)$ define the same curves in polar coordinates if $f_{1} (θ) = - f_{2} (θ + π)$ and use this to show that the following define the same conic section: $r = \frac{d e}{1 - e \cos θ}, r = \frac{- d e}{1 + e \cos θ}$

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Chapter 12, Problem 39CRE

Chapter 12, Problem 41CRE

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Chapter 12 Solutions

Calculus - Standalone book

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

Knowledge Booster

That r = f 1 ( θ ) and r = f 2 ( θ ) define the same curves in polar coordinates if f 1 ( θ ) = − f 2 ( θ + π ) and use this to show that the following define the same conic section : r = d e 1 − e cos θ , r = − d e 1 + e cos θ

Solution Summary: The author explains that r = f 1 ( ) and results in the same conic section.

That $r = f_{1} (θ)$ and $r = f_{2} (θ)$ define the same curves in polar coordinates if $f_{1} (θ) = - f_{2} (θ + π)$ and use this to show that the following define the same conic section: $r = \frac{d e}{1 - e \cos θ}, r = \frac{- d e}{1 + e \cos θ}$

Want to see the full answer?

Chapter 12 Solutions

That r = f 1 ( θ ) and r = f 2 ( θ ) define the same curves in polar coordinates if f 1 ( θ ) = − f 2 ( θ + π ) and use this to show that the following define the same conic section : r = d e 1 − e cos θ , r = − d e 1 + e cos θ

Solution Summary: The author explains that r = f 1 ( ) and results in the same conic section.

That r=f1(θ) and r=f2(θ) define the same curves in polar coordinates if f1(θ)=−f2(θ+π) and use this to show that the following define the same conic section:r=de1−ecosθ, r=−de1+ecosθ

Want to see the full answer?

Chapter 12 Solutions

That $r = f_{1} (θ)$ and $r = f_{2} (θ)$ define the same curves in polar coordinates if $f_{1} (θ) = - f_{2} (θ + π)$ and use this to show that the following define the same conic section: $r = \frac{d e}{1 - e \cos θ}, r = \frac{- d e}{1 + e \cos θ}$