Question

Let $z=(1-t)+i\sqrt{t^2+t+2}$ where t is real parameter then locus of z in the Gaussian plane is

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

Answer 1

Answer: (C)

hyperbola

$z=x+iy=(1-t)+i\sqrt{t^2+t+2}$ $\Rightarrow x=1-t$ and $y^2=t^2+1+2$ $\Rightarrow t+\frac{1}{2}=\frac{3}{2}+x$ and $y^2={(t+\frac{1}{2})}^2+\frac{7}{4}$ $\Rightarrow {(t+1/2)}^2={(\frac{3}{2}-x)}^2$ and $y^2={(t+1/2)}^2+\frac{7}{4}$$\Rightarrow y^2-\frac{7}{4}={(x-3/2)}^2$ $\Rightarrow -{(x-\frac{3}{2})}^2+y^2=\frac{7}{4}$ which represents a hyperbola.