Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 10 - Parametric and Polar Curves - 10.4 Conic Sections - 10.4 Exercises - Page 750: 27

Answer

$$\eqalign{ & {\text{Vertices: }}\left( { \pm 2,0} \right);\,\,{\text{Foci:}}\,\,\left( { \pm \sqrt 3 ,0} \right) \cr & {\text{major axis has length 4}} \cr & {\text{minor axis has length 2}} \cr} $$

Work Step by Step

$$\eqalign{ & \frac{{{x^2}}}{4} + {y^2} = 1 \cr & {\text{The equation has the standard form}} \cr & \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \cr & \frac{{{x^2}}}{4} + {y^2} = 1 \Rightarrow a = 2,b = 1 \cr & c = \sqrt {{a^2} - {b^2}} = \sqrt {{2^2} - {1^2}} = \sqrt 3 \cr & {\text{With}} \cr & {\text{Orientation of Major Axis Horizontal along the }}x{\text{ - axis, then}} \cr & {\text{Vertex }}\left( { - a,0} \right){\text{ and }}\left( {a,0} \right) \cr & {\text{Vertex}}\left( { - 2,0} \right){\text{ and }}\left( {2,0} \right) \cr & \cr & {\text{Foci }}\left( { - c,0} \right){\text{ and }}\left( {c,0} \right) \cr & {\text{Foci}}\left( { - \sqrt 3 ,0} \right){\text{ and }}\left( {\sqrt 3 ,0} \right) \cr & \cr & {\text{Length of the minor axis}} \cr & 2b = 2\left( 1 \right) = 2 \cr & {\text{Length of the major axis}} \cr & 2a = 2\left( 2 \right) = 4 \cr & \cr & {\text{Graph }} \cr} $$
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