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Math Help - Relationship between two complex numbers

  1. #1
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    Relationship between two complex numbers

    Given that z=w^2 where z=x+yi and w=u+vi, prove that:
    a) if u varies but v is constant, then the locus of P(x,y) is a parabola,
    b) if v varies and u is constant, then the locus of P is again a parabola.
    Show that these two parabolas have the same axis and focus.

    From the relation:
    x+yi=u^2-v^2+2uvi
    x=u^2-v^2
    y=2uv
    Which is the parametric equation for a parabola.
    When u varies, the parabola has focus (v,0) and axis y=0.
    When v varies, the parabola has focus (u,0) and axis y=0.

    The parabolas face in opposite directions, and both cross the y-axis, assuming that the constants are greater than 0. My problem is how do I show that they have the same focus. They do have the same axis.
    Thanks!
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  2. #2
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    Just to make it clear that "v is a constant", write c in place of v.

    x= u^2- v^2 and y= 2cu.

    From the second equation, u= \frac{y}{2c}

    Put that into the first equation: x= \frac{y^2}{4c^2}- c^2 which is the equation of a parabola with vertex at (-c^2, 0), axis along the x-axis, and opening to the right. You should have learned that the parabola y= \frac{1}{4c}x^2 has focus at (0, c). Adapt that to this problem.
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