Engineering Calculus Notes 142

Engineering Calculus Notes 142 - the asymptotes(the center...

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130 CHAPTER 2. CURVES x 2 a 2 y 2 b 2 = 1 x 2 a 2 y 2 b 2 = 1 x 2 a 2 y 2 b 2 = 0 F x = a/e c a b Figure 2.8: Hyperbolas and asymptotes Moving loci In the model equations we have obtained for parabolas, ellipses and hyperbolas in this section, the origin and the two coordinate axes play special roles with respect to the geometry of the locus. For the parabola given by Equation ( 2.10 ), the origin is the vertex , the point of closest approach to the directrix, and the y -axis is an axis of symmetry for the parabola, while the x -axis is a kind of boundary which the curve can touch but never crosses. For the ellipse given by Equation ( 2.13 ), the coordinate axes are both axes of symmetry, containing the major and minor axes, and the origin is their intersection (the center of the ellipse). For the hyperbola given by Equation ( 2.15 ), the coordinate axes are again both axes of symmetry, and the origin is their intersection, as well as the intersection of
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Unformatted text preview: the asymptotes (the center of the hyperbola). Suppose we want to move one of these loci to a new location: that is, we want to displace the locus (without rotation) so that the special point given by the origin for the model equation moves to ( α,β ). Any such motion is accomplished by replacing x with x plus a constant and y with y plus another constant inside the equation; we need to do this in such a way that substituting x = α and y = β into the new equation leads to the same calculation as substituting x = 0 and y = 0 into the old equation. It may seem wrong that this requires replacing x with x − α and y with y − β in the old equation; to convince ourselves that it is right, let us consider a few...
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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.

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