Question:
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is
(a) $|z|^{2}$
(b) $|\bar{z}|^{2}$
(c) $\frac{1}{2}|z|^{2}$
(d) none of these
Solution:
For any complex number $z,-i z$ represents complex number obtained by rotating $z$ clockwise by $\frac{\pi}{2}$ angle.
Hence, $z,-i z$ and $z+i z$ represents a right angled triangle with sides $z,-i z$ and hypotenus $z+i z$
$\therefore$ Area of triangle formed is
$=\frac{1}{2}|z \times(-i z)|$
$=\frac{1}{2}|i||z|^{2}$
$=\frac{1}{2}|z|^{2}$
Hence, the correct answer is option C.