The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is
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.

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