2.1. CONIC SECTIONS 127 • First, since x and y each enter Equation ( 2.13 ) only as their squares, replacing x with − x (or y with − y ) does not change the equation: this means the curve is invariant under refection across the y-axis . In particular, this gives us a second focus/directrix pair for the curve: F ( ae, 0) and x = a/e . • Second, it is clear that the ellipse is bounded: in fact the curve has x-intercepts ( ± a, 0) and y-intercepts (0 , ± b ). In the case a > b the distance 2 a ( resp . 2 b ) between the x-intercepts ( resp . y-intercepts) is called the major axis ( resp . minor axis ); the corresponding numbers a and b are the semimajor axis and the semiminor axis ,and the x-intercepts are sometimes called the vertices of the ellipse. When a < b , the names are interchanged. Equation ( 2.13 ) with b > a can be regarded as obtained from a version with b < a by interchanging x with y . Geometrically, this means that when b > a the foci are on the y-axis instead of the
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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.