University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 10 - Section 10.6 - Conics in Polar Coordinates - Exercises - Page 591: 28



Work Step by Step

The eccentricity of the ellipse $\dfrac{x^2}{p^2}+\dfrac{y^2}{q^2}=1$ when $p \gt q$ is given by: $e=\dfrac{\sqrt {p^2-q^2}}{p}$ The foci of the ellipse are: $(\pm pe,0)$ and the directrices are given as: $x=\pm \dfrac{p}{e}$ We are given that the vertices are: $(0, \pm 5)$ and $e=1.25$ so, $e=\dfrac{c}{a}=1.25\implies 5=\dfrac{5}{4}p \implies p=4$ Now, $q^2=c^2-p^2=25-16=9$ Thus, the equation of the ellipse becomes: $\dfrac{y^2}{16}-\dfrac{x^2}{9}=1$
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