Question

Let $z=1-t+i\sqrt{t^2+t+2}$, where $t$ is 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

The correct option is A a hyperbola

$z=1-t+i\sqrt{t^2+t+2}$
$x=1-t$ and $y=\sqrt{t^2+t+2}$
$\Rightarrow t=1-x$ and also $y^2=t^2+t+2$
$\Rightarrow y^2=(1-x{)}^2+1-x+2$
$\Rightarrow y^2=1+x^2-2x+1-x+2$
$\Rightarrow y^2=x^2-3x+4$ , which is a hyperbola