Explanation Results used: Polar equations of conic sections : Let d > 0. The conic section with a focus at the origin and eccentricity e has the polar equation r = e d 1 + e cos θ or r = e d 1 − e cos θ . The conci section with focus at the origin and eccentricity e has the polar equation r = e d 1 + e sin θ or r = e d 1 − e sin θ . If 0 < e <1, the conic section is an ellipse; if e = 1 ,it is a ellipse; and if e > 1 it is a hyperbola. The curves are defined over any interval in θ of length 2 π . Calculation: The given equation of the curve is r = 4 2 + cos θ . Rewrite the equation as r = 2 1 + 1 2 cos θ . Compare with r = e d 1 + e cos θ and obtain that e d = 2 and e = 1 2 . Therefore, obtain that d = 4 . As the term cos θ is in the equation the directrix are vertical. As e = 1 2 the curve is an ellipse. Therefore, one of the the directrix is x = 4 . As the directrix is vertical the vertex lies on the line which is parallel to the x -axis. So, to calculate the vertex set θ = 0 and θ = π in r = 2 1 + 1 2 cos θ . r = 2 1 + 1 2 cos 0 r = 2 1 + 1 2 r = 2 3 2 r = 4 3 Similarly, r = 2 1 + 1 2 cos π r = 2 1 + ( − 1 2 ) r = 2 1 2 r = 4 Therefore, the vertices are ( r 1 , θ 1 ) = ( 4 3 , 0 ) and ( r 2 , θ 2 ) = ( 4 , 0 ) In Cartesian coordinates the coordinates of vertices are ( r 1 cos θ 1 , r 1 sin θ 1 ) = ( 4 3 , 0 ) and ( r 2 cos θ 2 , r 2 sin θ 2 ) = ( − 4 , 0 ) . One of the foci is at origin and other focus will be at the distance which is same as the distance between one focus and the nearest vertex. Observe that the vertex ( 4 3 , 0 ) is nearest to the origin so the other focus will be at the distance of 4 3 from the vertex ( − 4 , 0 )

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

3rd Edition

ISBN: 9780135243435

Author: Bill Briggs

Publisher: Pearson Education (US)

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Chapter 12, Problem 63RE

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To graph: The curve r=42+cosθ and mention the vertices,foci,directrix and asymptotes ( if exists ) on the graph; verify the answer by using graphing utility.

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Chapter 12, Problem 62RE

Chapter 12, Problem 64RE

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Chapter 12 Solutions

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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To graph: The curve r = 4 2 + cos θ and mention the vertices,foci,directrix and asymptotes ( if exists ) on the graph; verify the answer by using graphing utility.

Solution Summary: The author calculates the polar equations of the conic sections by rewriting the equation as r=ed1+emathrmcostheta.

To graph: The curve r=42+cosθ and mention the vertices,foci,directrix and asymptotes ( if exists ) on the graph; verify the answer by using graphing utility.

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Chapter 12 Solutions