Question

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

• A. a hyperbola
• B. an ellipse
• C. a straight line
• D. None of these

Answer 1

Answer: (A)

$x+iy=1-t+i\sqrt{\left(t^2+t+2\right)}$
$\Rightarrow x=1-t$ 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.