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: 48

Answer

Equation of the ellipse: ${\left( {\frac{{x - 1}}{6}} \right)^2} + {\left( {\frac{y}{{3\sqrt 3 }}} \right)^2} = 1$

Work Step by Step

By Theorem 1, we have $P{F_1} + P{F_2} = 2a$. So, $a=6$. Since the foci are ${F_1} = \left( {4,0} \right)$ and ${F_2} = \left( { - 2,0} \right)$, the center of the ellipse is at $C = \left( {\frac{{4 - 2}}{2},0} \right) = \left( {1,0} \right)$. The ellipse centered at the origin would have equation ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = 1$. If the center is translated to $C=\left(1,0\right)$, the equation has the form: (1) ${\ \ \ }$ ${\left( {\frac{{x - 1}}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = 1$. Notice that $a$ and $b$ are unchanged after translation in this equation. So, we will work with the ellipse as if it is in the standard position to obtain $a$ and $b$. Then, we substitute the results in equation (1) to obtain the equation of the ellipse when it is centered at $C=\left(1,0\right)$. Since the foci are ${F_1} = \left( {4,0} \right)$ and ${F_2} = \left( { - 2,0} \right)$ and the center of the ellipse is at $C=\left(1,0\right)$. In standard position the foci are at $\left( { \pm c,0} \right) = \left( { \pm 3,0} \right)$. By Theorem 1, we have $c = \sqrt {{a^2} - {b^2}} $. So, $b = \sqrt {{a^2} - {c^2}} = \sqrt {36 - 9} = \sqrt {27} = 3\sqrt 3 $ Substituting $a$ and $b$ in equation (1) gives ${\left( {\frac{{x - 1}}{6}} \right)^2} + {\left( {\frac{y}{{3\sqrt 3 }}} \right)^2} = 1$
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