Engineering Calculus Notes 407

Engineering Calculus Notes 407 - loci described above, in...

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3.10. QUADRATIC CURVES AND SURFACES 395 If only one of the eigenvalues is nonzero, then λ 2 = 0; we can complete the square in the terms involving ξ 1 to write the equation in the form λ 1 p ξ 1 + α 2 λ 1 P 2 + βξ 2 = F + α 2 4 λ 1 ; The locus of this is a parabola as in Equation ( 3.36 ), but in the new coordinate system, displaced so the vertex is at ξ 1 = α/ 2 λ 1 2 = (4 λ 1 F + α 2 ) / 4 λ 1 . If both eigenvalues are nonzero, then we complete the square in the terms involving ξ 2 as well as in those involving ξ 1 to obtain λ 1 p ξ 1 + α 2 λ 1 P 2 + λ 2 p ξ 2 + β 2 λ 2 P 2 = F + α 2 4 λ 1 + β 2 4 λ 2 . This is Equation ( 3.38 ), Equation ( 3.39 ), or Equation ( 3.40 ), with x ( resp . y ) replaced by ξ 1 + α 2 λ 1 ( resp . ξ 2 + β 2 λ 2 ), and so its locus is one of the other
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Unformatted text preview: loci described above, in the new coordinate system, displaced so the origin moves to 1 = / 2 1 , 2 = / 2 2 . We illustrate with two examples. First, consider the curve 4 xy 6 x + 2 y = 4 . The quadratic form Q ( x,y ) = 4 xy has matrix representative A = [ Q ] = b 0 2 2 0 B with eigenvalues 1 = 2 , 2 = 1 and corresponding unit eigenvectors u 1 = p 1 2 , 1 2 P u 2 = p 1 2 , 1 2 P ; thus c = s = 1 2...
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