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 conic 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 = 12 3 − cos θ . Rewrite the equation as r = 4 1 − 1 3 cos θ . Compare the above equation with r = e d 1 − e cos θ and obtain that e d = 4 and e = 1 3 . Therefore, d = 12 . As the term cos θ is in the equation the directrix are vertical. As e = 1 3 the curve is an ellipse. As the equation has negative sign , one of the the directrix is x = − 12 . As the directrix is vertical the vertex lies on the line which is parallel to the x -axis. So to calculate the vertex θ = 0 and θ = π in r = 4 1 − 1 3 cos θ . r = 4 1 − 1 3 cos 0 r = 4 1 − 1 3 r = 4 2 3 r = 6 Similarly, r = 4 1 − 1 3 cos π r = 4 1 + 1 3 r = 4 4 3 r = 3 Therefore, the vertices are ( r 1 , θ 1 ) = ( 6 , 0 ) and ( r 2 , θ 2 ) = ( 3 , π ) . In Cartesian coordinates the coordinates of vertices are ( r 1 cos θ 1 , r 1 sin θ 1 ) = ( 6 , 0 ) and ( r 2 cos θ 2 , r 2 sin θ 2 ) = ( − 3 , 0 ) . One of the foci is at the 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 ( − 3 , 0 ) is nearest to the origin so the other focus will be at the distance of 3 from the vertex ( 6 , 0 )

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

See similar textbooks

Question

Chapter 12.4, Problem 62E

To determine

To graph: The curve r=123−cosθ and mention the vertices, foci, directrix and asymptotes ( if exists ) on the graph; verify the answer by using graphing utility.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.4, Problem 61E

Chapter 12.4, Problem 63E

Students have asked these similar questions

MindTap - Cemy X Answered: tat x A 26308049 × 10 EKU--SP 25:11 × E DNA Sequence x H. pylori index.html?elSBN=9780357038406&id=339416021&snapshotid=877369& NDTAP and the Derivative 41. 42. Answer 12 Ay 5 + -10-5 5 10 -5- f(x) = x +5 if x ≤ 0 -x²+5 if x > 0 to -5 5. 5 f(x) = |x − 1| MacBook Pro AA

Mind Tap - Cenxxx Answered: tat X A 26308049 × 10 EKU-- SP 25: X E DNA Sequence x H. pylor vo/index.html?elSBN=9780357038406&id=339416021&snapshotld=877369& MINDTAP its, and the Derivative 44. Answer 5 X -10-5 5 10 -5. f(x) = 2 + x +5 if x 0 3 4 f(x) = x² - 1 x+1 if x = -1 MacBook Pro G Search or type URL if x = -1 + AA a

Calculus lll May I please have an explanation of the multivariable chain rule in the example given? Thank you

Chapter 12 Solutions

Calculus, Single Variable: Early Transcendentals (3rd Edition)

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

Knowledge Booster

To graph: The curve r = 12 3 − 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 conic sections by comparing the equation with r=ed1+emathrmcostheta.

To graph: The curve r=123−cosθ and mention the vertices, foci, directrix and asymptotes ( if exists ) on the graph; verify the answer by using graphing utility.

Want to see the full answer?

Chapter 12 Solutions