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Math Help - Help needed with martices and conics

  1. #1
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    Question Help needed with martices and conics

    I'm not sure this is where I should put this, but I didn't see anywhere else:

    1)

    Using the determinant of a matrix representing a conic,
    [a b d]
    [b c e]
    [d e f]
    where the equation of the conic is
    ax^2=2bxy+cy^2+2dx+2ey+f=0
    and the determinant of the top left 2-by-2 matrix, how would you classify the conic?

    2) (Refering to the d and e of the previous question)

    Show that d^2+e^2 stays the same under a rotation of axes, in other words, the matrix is multiplied by
    [cos(z) -sin(z) 0]
    [sin(z) cos(z) 0]
    [0 0 1]

    Note: It's a 3x3 matrix

    Thanks in advance!
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  2. #2
    Super Member Rebesques's Avatar
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    1) Check the sign of the discriminant:

     b^2-ac  \begin{cases} <0: \mbox{ellipse}\cr =0: \mbox{parabola} \cr >0: \mbox{hyperbola} \end{cases}


    2) Let d',  \ e' be the respective coefficients after the rotation. Calculate to get d'=dcosz+esinz, \ d'=ecosz-dsinz. So d'^2+e'^2=d^2+e^2.
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