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Q.
Let $z=1-t+i \sqrt{\left(t^{2}+t+2\right)}$, where $t$ is a real parameter. The locus of $z$ in the Argand plane is
Complex Numbers and Quadratic Equations
Solution:
$x+i y=1-t+i \sqrt{\left(t^{2}+t+2\right)}$
$\Rightarrow x=1-t$ and $y=\sqrt{t^{2}+t+2}$
Eliminating t,
$y^{2}=t^{2}+t+2=(1-x)^{2}+1-x+2=\left(x-\frac{3}{2}\right)^{2}+\frac{7}{4}$
or $y^{2}-\left(x-\frac{3}{2}\right)^{2}=\frac{7}{4}$, which is a hyperbola.