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Q.
The area of the triangle whose vertices are the points represented by the complex number $z, i z$ and $z+i z$ is
Complex Numbers and Quadratic Equations
Solution:
Area of the triangle is given by
$\Delta=\frac{1}{4}\begin{vmatrix}z & \bar{z} & 1 \\ i z & -i \bar{z} & 1 \\ z+i z & \bar{z}-i \bar{z} & 1\end{vmatrix}$
Applying $R_3 \rightarrow R_3-R_1-R_2$, we get
$\Delta=\frac{1}{4}\begin{vmatrix}z & \bar{z} & 1 \\ i z & -i \bar{z} & 1 \\ 0 & 0 & -1\end{vmatrix}=\frac{1}{4}|(-1)\begin{vmatrix}z & \bar{z} \\ i z & -i \bar{z}\end{vmatrix} \mid$
$=\frac{1}{4}|(-1)(i) z \bar{z}| \begin{vmatrix}1 & 1 \\ 1 & -1\end{vmatrix}|=\frac{1}{4}z|^2(2)=\frac{1}{2}|z|^2$