Question

Area of the triangle in the argand diagram formed by the complex numbers $z,iz$ and$z-iz$, where $z=x+iy$ is:

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

Answer 1

Answer: (D)

$\frac{1}{2}|z{|}^2$

Let $A,B,C$ represent three complex number $z,iz$ and $z-iz$ on argand plane. Then $AB$ line will be parallel to $OC$ . So, Area of triangle $ABC$ will be equal to triangle $ABO$ as base $AB$ is same and point $O$ and $C$ lies on the line parallel to $AB$ .
Now, To find area of triangle $ABO$
$z$ and $iz$ are perpendicular to each other so area of the triangle will be $\frac{1}{2}base\times height$
Hence, Area $=\frac{1}{2}\times |z|\times |iz|$
Area$=\frac{|z{|}^2}{2}$