Question

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

The correct option is A a hyperbola

Let $z=x+iy$, then

$x+iy=1-t+i\sqrt{t^2+t+2}\Rightarrow x=1-t,y=\sqrt{t^2+t+2}$

Putting $t=1-x$ in the other relation, we get

$y^2=t^2+t+2={(1-x)}^2+1-x+2={(x-\frac{3}{2})}^2+\frac{7}{4}$

$\Rightarrow y^2-{(x-\frac{3}{4})}^2=\frac{7}{4},$ which is a hyperbola.