Engineering Calculus Notes 138

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 126 CHAPTER 2. CURVES Ellipses When e = 1, completing the square in Equation (2.8) and moving the y -axis to the right k/(1 − e2 ) units, we obtain (1 − e2 )x2 + y 2 = k2 e2 1 − e2 (2.11) as the equation of the conic section with eccentricity e = 1, directrix where x=− k , 1 − e2 and focus F( −ke2 ke2 , 0) = F ( , 0). e2 − 1 1 − e2 Noting that the x-coordinate of the focus is e2 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 − e2 This is positive provided 0 < e < 1, the case of the ellipse. If we divide both sides of Equation (2.11) by its right-hand side, we recognize the resulting coefficient for x2 as 1/a2 and get the equation x2 y2 +2 = 1. 2 a a (1 − e2 ) (2.12) In the case e < 1, we can define a new constant b by b=a 1 − e2 , so that (2.12) becomes x2 y 2 + 2 = 1. (2.13) a2 b This is the equation of the ellipse with focus F (−ae, 0) and directrix x = −a/e, where e= 1− b a 2 . Let us briefly note a few features of this curve: (2.14) ...
View Full Document

This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

Ask a homework question - tutors are online