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Math Help - complex no. + locus again

  1. #1
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    complex no. + locus again

    p294 q16 c
    given
    z^2-2z+k=0
    .......(*), where k is real, has no real roots.
    find , in terms of k, the squares of the roots of (*), expressing the answers in the form x+iy where x and real are real.
    as k varies, find the equation of the locus of the pts in the argand plane representing the squares of the roots of (*)

    my working:
    z = \frac {2\pm \sqrt{4-4k}}2 = 1 \pm \sqrt{1-k}
    square of the roots  = 2-k \pm 2\sqrt{k-1}i
    don't know how to find the locus. thanks.
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  2. #2
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    Quote Originally Posted by afeasfaerw23231233 View Post
    p294 q16 c
    given
    z^2-2z+k=0
    .......(*), where k is real, has no real roots.
    find , in terms of k, the squares of the roots of (*), expressing the answers in the form x+iy where x and real are real.
    as k varies, find the equation of the locus of the pts in the argand plane representing the squares of the roots of (*)

    my working:
    z = \frac {2\pm \sqrt{4-4k}}2 = 1 \pm \sqrt{1-k}
    square of the roots  = 2-k \pm 2\sqrt{k-1}i
    don't know how to find the locus. thanks.
    It's certainly a pleasure to see some working and a clear statement of where you're stuck.

    Note that since the roots are not real, k > 1.

    Real part: x = 2 - k, k > 1 .... (1)

    Imaginary part: y = \pm 2 \sqrt{k - 1}, k > 1 .... (2)

    From (1): k = 2 - x .... (3)

    Substitute (3) into (2) to get the Cartesian equation of the locus (it's a sideways parabola).
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