Explanation Given information: The polar equation, r = 8 4 + 8 cos θ Explanation: Theorem: The polar equation of conics with focus at the pole and directrix isperpendicular to the polar axis at a distance p units the right of the pole is r = e p 1 + e cos θ where e is the eccentricity of the conic. If e = 1 then conic is parabola and axis of symmetry is perpendicular to the directrix. If e > 1 then conic is hyperbola and axis of symmetry is perpendicular to the directrix. If e < 1 then conic is ellipse and axis of symmetry is perpendicular to the directrix. Convert the given equation r = 8 4 + 8 cos θ into r = e p 1 + e cos θ , Divide numerator and denominator on both sides by 4 , r = 2 1 + 2 cos θ Compare the given equation r = 2 1 + 2 cos θ with r = e p 1 + e cos θ , ⇒ e = 2 and e p = 2 . 2 ⋅ p = 2 By simplifying, ⇒ p = 1 . Since e = 2 > 1 , the conic is hyperbola with focus at the pole and directrix perpendicular to the polar axis at a distance p = 1 units the right of the pole. One focus of hyperbola is at the pole that means ( 0 , 0 ) . The Directrix of hyperbola is perpendicular to the polar axis at a distance p = 1 unit the right of the pole means x = 1 . The major axis is along the polar axis. To find the vertex of hyperbola, Let θ = 0 then r = 2 1 + 2 cos ( 0 ) = 2 1 + 2 ⋅ 1 = 2 3 ⇒ r = 2 3 Therefore, the vertex is ( 2 3 , 0 ) . Let θ = π then r = 2 1 + 2 cos ( π ) = 2 1 + 2 ( − 1 ) = 2 1 − 2 = − 2 1 r = − 2 Therefore, the vertex is ( − 2 , π ) . Thus, the vertex ( r , θ ) of the hyperbola are ( 2 3 , 0 ) and ( − 2 , π ) in polar co-ordinate. The vertex of hyperbola in rectangular coordinate are ( 2 3 , 0 ) and ( 2 , 0 ) . The center of hyperbola is the midpoint of the vertices ( 2 3 , 0 ) and ( 2 , 0 )

11th Edition

ISBN: 9780135189405

Author: Michael Sullivan

Publisher: PEARSON

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Chapter 10, Problem 29RE

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The conic represent by the polar equation and graph it where the polar equation is r=84+8cosθ.

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Chapter 10, Problem 28RE

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The conic represent by the polar equation and graph it where the polar equation is r = 8 4 + 8 cos θ .

Solution Summary: The author explains that the polar equation of conics with focus at the pole and directrix isperpendicular to a distance.

The conic represent by the polar equation and graph it where the polar equation is r=84+8cosθ.

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