Q249 / 270 · NDA 2017 (Phase 2) · Mathematics

Geometrically $\operatorname{Re}\left(z^2-i\right)=2$, where $i=\sqrt{-1}$ and $\operatorname{Re}$ is the real part, represents.

  • A
    circle
  • B
    ellipse
  • Correct answer
    C
    rectangular hyperbola
  • D
    parabola

Solution

$\operatorname{Re}\left(z^2-i\right)=2$ Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}$ Now, $z^2=(x+i y)^2=x^2+2 i x y-y^2$ $\begin{aligned}& z^2-i=x^2+2 i x y-y^2-i \\& =x^2-y^2+i(2 x y-1)\end{aligned}$ But, $\operatorname{Re}\left(z^2-i\right)=2$ i.e., $x^2-y^2=2$, which represents rectangular hyperbola.

Hence, the correct answer is option: 'C'.

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