Question

Area of the triangle in the argand diagram formed by the complex numbers $z, i z, z+i z$, where $z=x+i y$ is

• A. $|z|$
• B. $|z|^{2}$
• C. $2|z|^{2}$
• D. $\frac{1}{2}|z|^{2}$

Answer 1

Answer: (D)

since, $i z=z e^{i \frac{\pi}{2}}$
This implies that $i z$ is the vector obtained by rotating vector $z$ in anticlock wise direction through $90^{\circ}$
$\therefore OA \perp AB$

So, area of $\Delta OAB=\frac{1}{2} OA \times OB $
$=\frac{1}{2}|z||i z|=\frac{1}{2}|z|^{2}$