Q47 / 160 · AP EAMCET 2022 (08 Jul Shift 1) · Mathematics

Area of the triangle formed by the complex numbers $\mathrm{z}, \mathrm{iz}$ and $\mathrm{z}+\mathrm{iz}$ in the Argand diagram as vertices is

  • Correct answer
    A
    $\frac{1}{2} \cdot|z|^2$
  • B
    $\frac{1}{2} \cdot z^2$
  • C
    $\mathrm{z}^2$
  • D
    $|z|^2$

Solution

Let $Z=x+i y ; i z=-y+i x$ and $\mathrm{Z}+\mathrm{iz}=(\mathrm{x}-\mathrm{y})+\mathrm{i}(\mathrm{x}+\mathrm{y})$ $\therefore$ Area of triangle $\begin{aligned} & \varnothing=\frac{1}{2}\left|\begin{array}{ccc}x & y & 1 \\ x-y & x+y & 1 \\ -y & x & 1\end{array}\right| \\ & =\frac{1}{2}\left(x^2+y^2\right)=\frac{1}{2}|z|^2\end{aligned}$

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

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