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  4. Let z be a complex number such that z=1-t+ i √t2+t+2, where t ∈ R, then locus of z on the Argand plane is

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Q. Let $z$ be a complex number such that $z=1-t+$ $i \sqrt{t^2+t+2}$, where $t \in R$, then locus of $z$ on the Argand plane is

Complex Numbers and Quadratic Equations

Solution:

Let $z=x+i y$, then
$x=1-t, y=\sqrt{t^2+t+2} $
$\Rightarrow t=1-x \text { and } y^2=t^2+t+2=(t+1 / 2)^2+7 / 4 $
$\Rightarrow y^2=(x-3 / 2)^2+7 / 4$
This represents a hyperbola.