Complex geometry

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

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

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
You could make progress by considering interesting values (or limits) of t.