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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1715.0. "CMJ Problem 495" by RUSURE::EDP (Always mount a scratch monkey.) Thu Jan 28 1993 13:22

    Proposed by William R. Klinger, Taylor University, Upland, IN.
    
    Determine whether or not there is a cubic polynomial with three
    distinct real zeros a1, a2, a3 and two critical numbers c1, c2 (a1 < c1
    < a2 < c2 < a3) such that the segment from a1 to a2 is divided
    harmonically by c1 and c2.  [If the "cross-ratio"
    (c1-a1)(c2-a2)/(c1-a1)(c2-a1) of a1, a2, c1, and c2 is -1, then c1 and
    c2 are said to divide a1 and a2 harmonically.]
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1715.1NOT!GAUSS::ROTHGeometry is the real life!Fri Jan 29 1993 06:1852
>   Determine whether or not there is a cubic polynomial with three
>   distinct real zeros a1, a2, a3 and two critical numbers c1, c2 (a1 < c1
>   < a2 < c2 < a3) such that the segment from a1 to a2 is divided
>   harmonically by c1 and c2.  [If the "cross-ratio"
>   (c1-a1)(c2-a2)/(c1-a1)(c2-a1) of a1, a2, c1, and c2 is -1, then c1 and
>   c2 are said to divide a1 and a2 harmonically.]

   I believe there is a typo, the cross ratio should really be
   (c1-a1)(c2-a2)/(c1-a2)(c2-a1).

   The condition on the cross ratio is

   (c1-a1)(c2-a2) + (c1-a2)(c2-a1) = 0

   Let c1 = x0-dx and c2 = x0+dx

   (x0-a1-dx)(x0-a2+dx) + (x0-a2-dx)(x0-a1+dx) =

   2*(x0-a1)(x0-a2) - 2*dx^2 = 0					[1]

   Now, the cubic in terms of its roots is

   y = x^3 - (a1+a2+a3)*x^2 + (a1*a2 + a1*a3 + a2*a3) - a1*a2*a3

   y' = 3*x^2 - 2*(a1+a2*a3)*x + (a1*a2 + a1*a3 + a2*a3)		[2]

   The roots of y' are the c1 and c2, or x0-dx and x0+dx
   (where x0 happens to be the inflection point of the cubic)

   y' = 3*x^2 - 6*x0*x + 3*(x0^2 - dx^2)				[3]

   Comparing coefficients in [2] and [3]

   3*x0 = a1+a2+a3

   3*(x0^2 - dx^2) = a1*a2 + a1*a3 + a2*a3

   These can be subtituted in [1]

   (x0-a1)(x0-a2) - dx^2 = x0^2 - dx^2 - (a1+a2)*x0 + a1*a2 = 0

   3*(x0^2 - dx^2) - 3*(a1+a2)*x0 + 3*a1*a2 = 0

   a1*a2 + a1*a3 + a2*a3 - (a1+a2)*(a1+a2+a3) + 3*a1*a2 = 0

   -a1^2 + 2*a1*a2 - a2^2 = 0

   (a1-a2)^2 = 0

   But this contradicts that a1 and a2 are distinct.

    - Jim