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Engineering Calculus Notes 138

Engineering Calculus Notes 138 - 126 CHAPTER 2 CURVES...

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126 CHAPTER 2. CURVES Ellipses When e negationslash = 1, completing the square in Equation ( 2.8 ) and moving the y -axis to the right k/ (1 e 2 ) units, we obtain (1 e 2 ) x 2 + y 2 = k 2 e 2 1 e 2 (2.11) as the equation of the conic section with eccentricity e negationslash = 1, directrix where x = k 1 e 2 , and focus F ( ke 2 e 2 1 , 0) = F ( ke 2 1 e 2 , 0) . Noting that the x -coordinate of the focus is e 2 times the constant in the equation of the directrix, let us consider the case when the focus is at F ( ae, 0) and the directrix is x = a/e : that is, let us set a = ke 1 e 2 . This is positive provided 0 <e< 1, the case of the ellipse. If we divide both sides of Equation (
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