Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - 12.5 Conic Sections - Exercises - Page 636: 57


The conic section is an ellipse and the directrix is $x=8/3$.

Work Step by Step

Converting the given equation to the standard form $$ r=\frac{e d}{1+e \cos \theta}. $$ We get $$ r=\frac{8}{4+ 3\cos \theta}=\frac{2}{1+(3/4) \cos \theta}. $$ Thus $ e=\frac{3}{4}.$ Since $e\lt 1$ then the conic section is an ellipse. To find the directrix, we find $d$ first. Since $ed=2$ and $e=3/4$ then $d=8/3$ and hence the directrix is $x=8/3$.
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