Question

For the real parameter t, the locus of the complex number $z=\left(1-t^2\right)+i\sqrt{1+t^2}$ in the complex plane is

• A. an ellipse
• B. a parabola
• C. a circle
• D. a hyperbola

Answer 1

Answer: (B)

Let $z=x+iy$ Given, $z=\left(1-t^2\right)+i\sqrt{1+t^2}$
$\therefore x+iy=\left(1-t^2\right)+i\sqrt{1+t^2}$
On equating real and imaginary parts, we get
$x=1-t^2$ and $y=\sqrt{1+t^2}$
$\therefore x+y^2=1-t^2+\left(1+t^2\right)$
$\Rightarrow x+y^2=2$
$\Rightarrow y^2=-(x-2)$
Hence, it represents a parabola.