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Complex geometry

  1. Apr 27, 2016 #1
    1. The problem statement, all variables and given/known data
    If ##α, β, γ, δ## are four complex numbers such that ##\dfrac{γ}{δ}## is real and ##αδ - βγ ≠ 0##, then ##z = \dfrac{α + βt}{γ + δt} , t \in ℝ## represents a
    (A) circle
    (B) parabola
    (C) ellipse
    (D) straight line

    2. Relevant equations


    3. The attempt at a solution
    Eqn of circle is ##|z - z_0| = k##, ellipse is ##|z - z_1| + |z - z_2| = k, |z_1 - z_2| < k##, straight line is ##\arg(z - z_0) = k## and not sure how I'd represent a parabola's complex equation, though it'd be something like distance from a straight line is equal to distance from a point, so maybe something like ## |z - z_0| =\dfrac{ |\bar{a}z + a\bar{z} + b|}{2|a|}##
    Since ##\dfrac{γ}{δ}## is purely real ##\dfrac{γ}{δ} = \dfrac{\bar{γ}}{\bar{δ}}##
    Beyond this, I'm hopelessly clueless. Please help.
     
  2. jcsd
  3. Apr 27, 2016 #2

    haruspex

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    You could make progress by considering interesting values (or limits) of t.
     
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