Complex geometry

  • Thread starter erisedk
  • Start date
  • #1
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Homework Statement


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

Homework Equations




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.
 

Answers and Replies

  • #2
haruspex
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You could make progress by considering interesting values (or limits) of t.
 

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