# 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 $\dfrac{γ}{δ} = \dfrac{\bar{γ}}{\bar{δ}}$