Engineering Calculus Notes 139

Engineering Calculus Notes 139 - 127 2.1 CONIC SECTIONS...

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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 reflection 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 x -axis (and the directrices are horizontal). Third, if we know the semimajor axis a and the semiminor axis b , then the distance from the origin to the two foci is given by the
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