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 $z=x+i y$ Given, $z=\left(1-t^{2}\right)+i \sqrt{1+t^{2}}$
$\therefore x+i y=\left(1-t^{2}\right)+i \sqrt{1+t^{2}}$
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.