Engineering Calculus Notes 405

Engineering Calculus Notes 405 - the conic sections is the...

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3.10. QUADRATIC CURVES AND SURFACES 393 the linear terms are present, one needs to complete the square(s) to determine whether the given equation came from one of the type above, or one with zero or negative right-hand side. Hyperbolas: the model equations x 2 a 2 y 2 b 2 = ± 1 (3.39) for a hyperbola centered at the origin and symmetry about both coordinate axes corresponds to B = 0, A and C of opposite signs, F n = 0, and D = 0 = E . When F = 0 but the other conditions remain, we have the equation x 2 a 2 y 2 b 2 = 0 (3.40) which determines a pair of lines, the asymptotes of the hyperbolas with F n = 0. As before, moving the center introduces linear terms, but completing the square is needed to decide whether an equation with either D or E (or both) nonzero corresponds to a hyperbola or a pair of asymptotes. In eFect, the list above (with some obvious additional degenerate cases) takes care of all versions of Equation ( 3.35 ) in which B = 0. Unfortunately, when B n = 0 there is no quick and easy way to determine which, if any, of
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Unformatted text preview: the conic sections is the locus. However, if it is, it must arise from rotation of one of the model versions above. We will see that the locus of every instance of Equation ( 3.35 ) with not all leading terms zero has a locus ±tting one of these descriptions (with diFerent centers, foci and directrices), or a degenerate locus (line, point or empty set). To this end, we shall start from Equation ( 3.35 ) and show that in an appropriate coordinate system the equation ±ts one of the molds above. Let us denote the polynomial on the left side of Equation ( 3.35 ) by p ( x,y ): p ( x,y ) = Ax 2 + Bxy + Cy 2 + Dx + Ey. Assuming they don’t all vanish, the leading terms de±ne a quadratic form Q ( x,y ) = Ax 2 + Bxy + Cy 2 with matrix representative A = [ Q ] = b A B/ 2 B/ 2 C B ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Spring '08 term at University of Florida.

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