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: 37


Ellipse: - the vertices are $(1\pm \frac{5}{2}, \frac{1}{5})$, $(1, \frac{1}{5}\pm1)$ - the foci are $(1\pm \sqrt{21/4},1/5)$ - the center is $(1,1/5)$

Work Step by Step

By completing the square, we can write the equation $$ 4 x^{2}+25 y^{2}-8 x-10 y=20 $$ in the form $$ 4 (x-1 )^2+25 (y-\frac{1}{5})^2=25. $$ Then we get $$ \left(\frac{x-1}{ 5/2}\right)^{2}+\left(\frac{y-\frac{1}{5}}{1}\right)^{2}=1 $$ which is an ellipse with $a=5/2, b=1$ and hence $ c=\sqrt{a^2-b^2}=\sqrt{21/4} $ So, we have: - the vertices are $(1\pm \frac{5}{2}, \frac{1}{5})$, $(1, \frac{1}{5}\pm1)$ - the foci are $(1\pm \sqrt{21/4},1/5)$ - the center is $(1,1/5)$
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