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) $|\overline{z}{|}^2$

(c) $\frac{1}{2}{\left|z\right|}^2$

(d) none of these

Answer 1

For any complex number z, –iz represents complex number obtained by rotating z clockwise by $\frac{\mathrm{\pi }}{2}$ angle.
Hence, z, –iz and z + iz represents a right angled triangle with sides z, –iz and hypotenus z + iz
∴ Area of triangle formed is
$$\begin{gathered} =\frac{1}{2}\left|z\times \left(-iz\right)\right| \\ =\frac{1}{2}\left|i\right|{\left|z\right|}^2 \\ =\frac{1}{2}{\left|z\right|}^2 \end{gathered}$$
Hence, the correct answer is option C.