Question

Let $z=1-t+i\sqrt{t^2+t+2}$, where $t$ is a real parameter. The locus of $z$ in Argand plane is

• A. a hyperbola
• B. an ellipse
• C. a straight line
• D. a circle

Answer 1

The correct option is A a hyperbola

Given : $z=1-t+i\sqrt{t^2+t+2}$
Let $z=x+iy$, then
$\Rightarrow x=1-t,y=\sqrt{t^2+t+2}$
Eliminating $t$,
$y^2=t^2+t+2\Rightarrow y^2=(1-x{)}^2+1-x+2\Rightarrow y^2=x^2-3x+4\Rightarrow y^2={(x-\frac{3}{2})}^2+\frac{7}{4}\therefore \frac{y^2}{7/4}-\frac{{(x-\frac{3}{2})}^2}{7/4}=1$

This is a equation of hyperbola.