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Thread: Relationship between two complex numbers

  1. #1
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    Relationship between two complex numbers

    Given that $\displaystyle z=w^2$ where $\displaystyle z=x+yi$ and $\displaystyle w=u+vi$, prove that:
    a) if u varies but v is constant, then the locus of $\displaystyle P(x,y)$ 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:
    $\displaystyle x+yi=u^2-v^2+2uvi$
    $\displaystyle x=u^2-v^2$
    $\displaystyle y=2uv$
    Which is the parametric equation for a parabola.
    When u varies, the parabola has focus $\displaystyle (v,0)$ and axis $\displaystyle y=0$.
    When v varies, the parabola has focus $\displaystyle (u,0)$ and axis $\displaystyle y=0$.

    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
    MHF Contributor

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    Just to make it clear that "v is a constant", write c in place of v.

    $\displaystyle x= u^2- v^2$ and $\displaystyle y= 2cu$.

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

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