The area of the triangle on the argand diagram formed by the complex numbers, $z, i z$ and $z+i z$ i

Question

The area of the triangle on the argand diagram formed by the complex numbers, $z, i z$ and $z+i z$ is $n 	imes|z|^{2}$. n=____?

Professor PC Thomas Classes · Answer

Let $A=z=x+i y, B=i z=-y+i x$\ $C=z+i z=(x-y)+i(x+y)$\ Now area of $	riangle A B C=\frac{1}{2}\left|\begin{array}{ccc}x & y & 1 \ -y & x & 1 \ x-y & x+y & 1\end{array}ight|$ Operate $R_{2} ightarrow R_{2}-R_{1}, R_{3} ightarrow R_{3}-R_{1}$\ $	herefore \quad$ Area of $	riangle A B C=\frac{1}{2}\left|\begin{array}{ccc}x & y & 1 \ -y-x & x-y & 0 \ -y & x & 0\end{array}ight|$\ $=\frac{1}{2}[-x(y+x)+y(x-y)]$\ $=\frac{1}{2}\left[-x y-x^{2}+x y-y^{2}ight]$\ $=\frac{1}{2}\left[-\left(x^{2}+y^{2}ight)ight]=\frac{1}{2}|z|^{2}$\ $	herefore \quad$ Area of triangle $A B C=\frac{1}{2}|z|^{2}$\